The Third Road to Quantum Gravity

[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.

 

[Careful]

**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. **

I did not mean to define what a proposition is, I wanted to speak about a set of propositions such as : ``the moon is there, or the Earth is round, etc... ´´

**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.**

I KNOW that, but (a) it is always GOOD to give an example which is understandable for everyone (you have to learn to communicate an idea intuitively, and many mathematicians often can't - they are stuck in their details) (b) you seem not to appreciate my comments that deviating from manifoldness too violently highly likely leads to nonrenormalizable theories.


**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. **


So what do you start with (we need to know what we are talking about!)?? The use of energy scales in a fundamental theory is IMO highly anti relativistic, but ok, the high energy community would back you up here.

**Physically, energies are no different to quantum numbers: they need to be looked at in the context of the experiment.**

So, you stick to the reduction postulate in QM? Right?


**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.**

That is a difficult question REGARDLESS of categorical considerations.

**Hence the question what category do you restrict to is completely meaningless.**

It is not ! If you do not do that, it is impossible for you to define a controllable dynamics! I would appreciate it if you would comment on my other remarks too and not only select those which are specifically category theory oriented.

 

[Kea]

I did not mean to define what a proposition is, I wanted to speak about a set of propositions such as: "the moon is there" or "the Earth is round" etc...

Yes, but the idea that physical propositions can form a fixed set is IMHO exactly the problem. The notion of a fixed classical reality is tantamount to the inclusion of a universal observer in one's framework. In the reduction of the general picture to ordinary QM one must of course recover some such description, but I believe it is a prejudice that needs doing away with.

...it is always GOOD to give an example which is understandable for everyone.

OK. Fair enough.

...you seem not to appreciate my comments that deviating from manifoldness too violently highly likely leads to nonrenormalizable theories.

Except that, to my knowledge, the best understanding of renormalisation that we have comes from the current Connes, Marcolli, Kreimer et al work, which is beginning to use motivic cohomology and such things rather heavily...

So what do you start with (we need to know what we are talking about!)?

I guess what you want to know is: what higher dimensional categories should one operate into be able to do either (a) GR or (b) the standard model. For the latter it would seem a suitable choice of Gray categories (with the ability to do quantum logic), such as a rich form of Vect, would do the trick.

So, you stick to the reduction postulate in QM? Right?

Well, not exactly. Although the only way I know how to speak about observables is in a language that sounds like QM, it is quite certain that the topos-like axiomatics are naturally written in a form that only reduces to the usual case on the choice of a particular model.

The interplay between theories and models is an important aspect of the logical point of view, and one of the main reasons that I make the claim that categories are much more than an organisational tool.

 

[Careful]

**Yes, but the idea that physical propositions can form a fixed set is IMHO exactly the problem. The notion of a fixed classical reality is tantamount to the inclusion of a universal observer in one's framework. **

AH, but that depends upon what you want to do with QM (and that is the whole crux of the story). I mentioned previously that for a classical theory, there is no problem, the set of physical propositions we can make is DYNAMICAL (I think I explicitely mentioned this already in post 97). Example: put some differential equation in your computer with exotic intial conditions (possibly with boundary conditions). Suddenly you notice that very complex patterns form in time for this particular solution mankind has never seen before. At that moment you learn something more and your set of propositions gets enlarged. All you need for that is a classical theory of the brain which allows for pattern recognition and an (classical) arrow of time.

Now, in ordinary QM you are screwed, the propositions are merely put in by the experimentator and not dynamically generated at all. So, this leaves a few possibilities :

(a) you introduce a double world picture, both with its own dynamics and a principle which interrelates them. This is for example done in the work of Aerts et al (based upon work of Piron). He considers quantum words embedded in different classical entities (the problem is that there is no dynamics yet). The kinematical setting of this work is very abstract but concrete (they work in a specific category as you would call it)
(b) Everything is quantum but then my guess is that you have to make QM nonlinear to make sense out of the macroworld (and as such out of a dynamical set of propositions)
(c) Everything is classical and the Schroedinger equation is nothing but a statistical divise containing macroscopic parameters (like mass and charge) to compute outcomes of experiments. Therefore, we have to look for an underlying deterministic mechanics (my preferred approach)

If you have something to add, please go. You see: you can adress this problem within a fixed ``master´´ category (you just have to choose it big enough so that it suits your desires).

**Except that, to my knowledge, the best understanding of renormalisation that we have comes from the current Connes, Marcolli, Kreimer et al work, which is beginning to use motivic cohomology and such things rather heavily...**

I don't know that and I would like to hear what NEW insight is gained vis a vis the methods physicists normally use.

**I guess what you want to know is: what higher dimensional categories should one operate into be able to do either (a) GR or (b) the standard model. For the latter it would seem a suitable choice of Gray categories (with the ability to do quantum logic), such as a rich form of Vect, would do the trick. **

Sure, everyone would like to know this ! So, I suggest that you concentrate on explaining the virtues of these two categories.


**The interplay between theories and models is an important aspect of the logical point of view, and one of the main reasons that I make the claim that categories are much more than an organisational tool. **

Hmmm, but it are in the end only the models which form the *physical* theory. So your level of abstraction is good for mathematics, but in the end physics concentrates itself on a specific kinematics endowed with a specific DYNAMICS.

Cheers,

Careful

 

[Kea]

So, this leaves a few possibilities :
(a) you introduce a double world picture, both with its own dynamics and a principle which interrelates them. This is for example done in the work of Aerts et al (based upon work of Piron). He considers quantum words embedded in different classical entities (the problem is that there is no dynamics yet). The kinematical setting of this work is very abstract but concrete (they work in a specific category as you would call it)
(b) Everything is quantum but then my guess is that you have to make QM nonlinear to make sense out of the macroworld (and as such out of a dynamical set of propositions)
(c) Everything is classical and the Schroedinger equation is nothing but a statistical divise containing macroscopic parameters (like mass and charge) to compute outcomes of experiments. Therefore, we have to look for an underlying deterministic mechanics (my preferred approach)

My preference is for a mixture of (a) and (b). My work is probably most closely related to (b). Piron's ideas have led many people to topos theory. Quite frankly, as a physicist, I don't see how you can take (c) seriously.

I don't know that and I would like to hear what NEW insight is gained vis a vis the methods physicists normally use.

The renormalisation studies are about the methods physicists normally use. Sure, it's very mathematical, and I can well understand why any physicist would be reluctant to touch it...but I reserve the right to burden myself with any mathematics that I have good reason to think supports this approach.

So, I suggest that you concentrate on explaining the virtues of these two categories.

That is, naturally, the long term goal of this and some other threads. If I could do this in the next 5 minutes, I would be busy calculating things and not wasting my time here. Moreover, PF has no facility for drawing lots of diagrams.

Hmmm, but it are in the end only the models which form the *physical* theory.

Well, time will tell, won't it? Careful, you have not given us a single solid argument to deter people from this approach. Once again, I urge you to begin another thread, introducing us to your own ideas.

 

[Careful]

**My preference is for a mixture of (a) and (b). My work is probably most closely related to (b). Piron's ideas have led many people to topos theory. Quite frankly, as a physicist, I don't see how you can take (c) seriously.***


My God, you are completely unaware of the fact that the best thinkers in physics have mostly taken (c) seriously (and continue to do so). To name a few: Einstein, de Broglie, 't Hooft, A.O Barut (partly) etcetera... . ¨I want to stress that (c) is a logical possibiltiy which has NOT been experimentally refuted. Moreover, (a) as I said is really *inexistant* in the sense that there is no DYNAMICS yet (although this approach goes along for some twenty years too no?) (b) has been the subject of intense research without any conclusive outcome so far. Considering (c) is usually frowned upon albeit there exist already some partial results in that direction (SED, Self field approach etcetera)! These theories partly reproduce in a (semi) classical way quantum results (so they left the kinematical stage already for a some 25 years) and have a clear (classical) ontology.


**The renormalisation studies are about the methods physicists normally use. Sure, it's very mathematical, and I can well understand why any physicist would be reluctant to touch it...but I reserve the right to burden myself with any mathematics that I have good reason to think ssupports this approach. **

So, you cannot tell us why it brings us anything extra while you used this argument to ``counter´´ my objection that if you make the category too wild, the dynamics you are interested in is probably nonrenormalizable. There is serious evidence for this claim: check out the literature in dynamical triangulations to start with. Moroever such debating tricks are not serious : you should consider going into politics since you clearly do not like to answer real objections.


**That is, naturally, the long term goal of this and some other threads. If I could do this in the next 5 minutes, I would be busy calculating things and not wasting my time here. Moreover, PF has no facility for drawing lots of diagrams. **

You can still explain in words...

**Well, time will tell, won't it? Careful, you have not given us a single solid argument to deter people from this approach. Once again, I urge you to begin another thread, introducing us to your own ideas.
**

O yeh, I have given pleanty of them: (a) it did not bring anything new yet (b) I did not hear yet any argument why it should contribute one day to physics (on the contrary, there is evidence that you have to keep your kinematical structure under control - but of course you are completely ignoring these facts or are simply not aware of them) (c) the examples you gave us are entirely *kinematical* and avoid as such the real problem: that is the dynamics (d) the good inventions in physics NEVER emerged from a twisted desire to abstraction, but always were the result of deep, albeit relatively simple ideas and calculations.

So, IMO you are a mathematician who is on a promotional tour trying to link her business to ``physics´´. Perhaps I am wrong and you can still present us some real physics = kinematics + dynamics.