Exploring Careful's Reflections on QG and Category Theory

[Kea]

So that we do not disturb the Freidel et al thread, I will start another based on Careful's remarks:

The reason why I am asking these similar questions over and over again when I hear these cries is because I believe a deeper failure in our theories to be responsible for the problems of QG while such lines of thought merely reflect a technical issue. Therefore, I would welcome any insight which shows me otherwise...

First comment: if you had made any effort whatsoever to look at category theory over the last few months you would realize that the assumption that it "isn't deep enough" might be a little premature. The reason for our anger is that it should be perfectly obvious that these technical results we are discussing fit into a larger framework.

Re gauge groups: in the commutative case, at the heart of topos theory is a categorical equivalence known as Stone duality. One form of this is familiar to you as the ordinary Fourier transform. The equivalence singles out $U(1)$ as a special object amongst commutative spaces because it looks like both a space and something dual to this. The fact that Barrett's ideas appear prominently in Baratin-Freidel is no coincidence, because Barrett specialises in studying q-deformed Fourier transforms.

What one needs then is a noncommutative analogue...and the claim is that the whole of Connes' program of NCG is not deep enough...but pure category theory is. The secret is cohomology. Some time in the 60s or 70s there was a split of Grothendieck's ideas (arising out of Weil conjectures and lots of fancy maths) into the Algebraic Geometry camp and the pure category theory camp. It is now clear to mathematicians that the latter is more powerful. There are also thousands upon thousands of expository articles on this history, some of them quite readable.

Something like q-deformed $SU(2) \times SU(2)$ is a natural candidate for a noncommutative (quaternionic) self-dual object, in the sense above. I know of no proof of this...that would involve a proper understanding of higher Stone duality. Suffice it to say that these questions are equivalent to answering questions like the Riemann hypothesis...a generalisation of which would follow from an analysis of this duality. Take a look at the papers on Feynman diagrams and MZVs by Kreimer, Broadhurst and others. They are very numerical and easy to read.

I hope that is a start.

P.S. I am the barmaid.

 

[Careful]

***
First comment: if you had made any effort whatsoever to look at category theory over the last few months you would realize that the assumption that it "isn't deep enough" might be a little premature. The reason for our anger is that it should be perfectly obvious that these technical results we are discussing fit into a larger framework. **

No, why would I have done that? Why do people always assume that everyone reads the thing they are doing (while they don't do it vice versa themselves) ? I see you are throwing many mathematical concepts at us (me), some of which I can follow only intuitively. Ok, let's start like this : imagine that I am an old grumpy physicist sipping from a dark beer and badly dressed , basically I don't care too much about deformed groups yet (seems to be a fancy way to sneak in a cutoff), neither do I care about the Riemann hypothesis, fancy dualities and so on. Neither am I a fan of Baratin, Freidel and all this LQG related stuff, *but* I would become more excited when someone can shed deeper light upon the physical issues at hand, give an intuitive picture why what he/she is doing is promising from that point of view. Then, I would be prepared to take a look at the math. So far, even Hurkyl who seems more of an enthousiast declared that it went about TQFT on a fixed background only - now in that direction abstract ideas have been developped enough, so in what sense does what you do surpass the fixed background as well as previous work on it (Markopoulou etc...); what is the new *physical* input ?? And sorry, if you read the Baez paper as it stands from an outsider's point of view, then you would also ask where the beef really is (you might also adress my particular objection on that construction).

Careful

 

[Kea]

...imagine that I am an old grumpy physicist sipping from a dark beer and badly dressed...

Grumpy old physicists will never be convinced. Only young physicists matter.

OK. Let's say for the moment that our particular emphasis on locality has a similar look to it to the intuition from AQFT. Of course backgrounds are dreadful because the observation of a whole lot of stuff (locally) is complex and cannot possibly be fundamental. Now, although you continually reject QM, I hope you are not so hasty as to reject localisation in AQFT. The question is: how do we get rid of the Minkowski background here? That's where the topos theory comes in, for starters. After thinking about this for a while...say 20 years...and after wondering how a Machian principle can possibly be written down...one can't help getting bogged down by some pretty hairy mathematics. Anyway...

What is my notion of time? I don't care about yours to start with, because you live in your world. All I know is that if I want to decide what epoch of a GR universe I live in, something I will do is measure the CMBR. But a high energy ant living on Earth would measure something different, and come to its own conclusions about what epoch it inhabited. So, first point, clearly the classical notion of time is not sufficient to cope with plausible experiments. But BI notions of time must of course be postSM as well as postGR. How on Earth can one mathematically do this? Category theory can do it.

You may object that I'm thinking quantum mechanically, but all I'm saying is that we should start thinking about what we measure. Heisenberg was actually led to the UP by looking at cloud chamber tracks and thinking about what Einstein had told him.

 

[Careful]

**Grumpy old physicists will never be convinced. Only young physicists matter. **

Because it is easier to convince them I guess...

**
OK. Let's say for the moment that our particular emphasis on locality has a similar look to it to the intuition from AQFT. Of course backgrounds are dreadful because the observation of a whole lot of stuff (locally) is complex and cannot possibly be fundamental. **

In physics you always have to draw some ``reality'' conclusions, in QM measurement is real, but measurement should have the same status as the thing being measured (that is one problem). So, as a realist I can posit hidden variables and declare them to be real, what we measure is simply some coarse grained byproduct, I don't need to go over to QM or anything whatsoever to satisfy that requirement.

**
Now, although you continually reject QM, I hope you are not so hasty as to reject localisation in AQFT. **

I don't reject the Dirac field (the KG field has more problems) and the non-relativistic Schroedinger wave, I do reject the operational definition of measurement as being fundamental though. I am not hasty in rejecting anything... it took me 10 years to say farewell to some part of QM.

**
The question is: how do we get rid of the Minkowski background here? That's where the topos theory comes in, for starters. After thinking about this for a while...say 20 years...and after wondering how a Machian principle can possibly be written down...one can't help getting bogged down by some pretty hairy mathematics. Anyway... **

Ok, so here you could expand : (a) what do you mean with the Machian principle ? Is it reasonable to expect it to be a part of physics? On another thread, I said that Mach's principle when you interpret it very strongly is a form of intelligent self organisation in nature. As local realist, I see this as entities keeping lists of previous interactions with their neighbours and adapting their ``weight'' in the fundamental field equations. Now, you can go as far as you want to with this, and what we call conspiracy in *some interpretation of* EPR could be very natural in such context. But, on the other hand, one realizes immediately that one can virtually do anything with ``intelligent'' design... so where to start and where to end ?
(b) second, to get rid of the minkowski background, why would you want to do that (if one were to look at it as a mere technical devise)? By saying so, you immediately seem to jump to the conclusion that the failure of perturbative QG is a technical issue (the wrong split : free part + interactions) and not of deeper physical meaning (such as string theorists think to some extend).

**
What is my notion of time? I don't care about yours to start with, because you live in your world. All I know is that if I want to decide what epoch of a GR universe I live in, something I will do is measure the CMBR. But a high energy ant living on Earth would measure something different, and come to its own conclusions about what epoch it inhabited. So, first point, clearly the classical notion of time is not sufficient to cope with plausible experiments. **

Huh, why do you say that? I can have a universal notion of time which indicates when the ant and I are simultaneously ``alive'', how the ant *percieves* time is an entirely different matter - depends upon his internal biological clock, number of hartbeats per minut and so on.
You seem to say that a subjective perception of the world implies a subjective physics (Rovelli style).

** But BI notions of time must of course be postSM as well as postGR. How on Earth can one mathematically do this? Category theory can do it.
**
I don't see that, would you care to define a BI notion of time ?? Even quantum mechanically, I could take time to be an extra parameter.

**
You may object that I'm thinking quantum mechanically, but all I'm saying is that we should start thinking about what we measure. Heisenberg was actually led to the UP by looking at cloud chamber tracks and thinking about what Einstein had told him. **
**

I don't know why you think that classical physicists don't think about what we measure. We do see measurement as a local physical process with definite outcome though -which is actually not in contradiction to experiment at all and was Schrodingers (and Einstein's) view upon QM.

 

[Kea]

The crux of the disagreement, as usual, appears to be your unwillingness to consider a measurement principle as fundamental. Now, I haven't actually given you a technical statement of this device because it isn't possible to understand it without quite a lot of category theory.

(a)what do you mean with the Machian principle? Is it reasonable to expect it to be a part of physics?

I interpret it after taking on board an idea for measurement in terms of interacting toposes. This means that the Machian duality between local buckets and far-away stuff is a statement about propositions. I will only consider it a part of physics when we have properly derived the SM parameters.

As local realist, I see this as entities keeping lists of previous interactions with their neighbours... on the other hand, one realizes immediately that one can virtually do anything with "intelligent'' design...so where to start and where to end?

Your thinking is too ontological. That may suffice for your approach, but it leads to inconsistencies in ours. I challenge you to answer Bee's questions in a self-consistent manner.

(b)second, to get rid of the minkowski background, why would you want to do that?

Because it is not the correct starting point from the principles being used.

You seem to say that a subjective perception of the world implies a subjective physics (Rovelli style).

Basically, yes. Rovelli's ideas will work as a first approximation.

Even quantum mechanically, I could take time to be an extra parameter.

Yes, but we are not doing QM.

...which is actually not in contradiction to experiment at all...

Again, I challenge you to explain ALL the cosmological problems with local realism. No one here ever said that there was nothing to your approach.

 

[Careful]

** The crux of the disagreement, as usual, appears to be your unwillingness to consider a measurement principle as fundamental. **

Unwillingness ?? Why as a rational person would you accept such thing when it turns out not to be necessary at all (yet).

**
Now, I haven't actually given you a technical statement of this device because it isn't possible to understand it without quite a lot of category theory.**

Barmaid !? Give it a shot, we might actually learn *both* from it.

**
I interpret it after taking on board an idea for measurement in terms of interacting toposes. This means that the Machian duality between local buckets and far-away stuff is a statement about propositions. I will only consider it a part of physics when we have properly derived the SM parameters. **

Ok, it sounds reasonable to consider it only at that point (in your line of thought). But you still didn't tell us what you understand under Machian duality in terms of measurements.

**
Your thinking is too ontological. That may suffice for your approach, but it leads to contradictions in ours. I challenge you to answer Bee's questions in a self-consistent manner. **

I did not read them, but I guess it is the usual lore about the cosmological constant, dark matter, what happens next to the big bang, explanation for CMB and so on. Now, the cosmological constant is by far the most puzzling and probably requires new physics. Dark matter models such as Mond are being developped by cosmologists, they are entirely classical and if you want to have a feeling for the quantum corrections - simply take the classical solution as a background and calculate the leading order ``graviton'' corrections. What happens next to the big bang ? Relativity is incorrect at the Planck scale and is only an effective low energy approximation to our word - the Planck scale dynamics can assumed to be deterministic. I believe these cosmological questions can be answered using classical mechanisms only : singularity avoidance through a bounce in the isotropic homogeneous sector can already be established in a simple model such as gravity coupled to *one* (so no conflict with local realism here !) complex KG field in a Bohm de Broglie approach to QG (now I make tnn happy).
There is plenty of evidence that the big questions of QG can be adressed in much ``simpler'' models than people think - if you want to I will give papers. Therefore, I am mainly concerned about recovering quantum predictions where they have been tested - as I should ! And Barut got a long, long way in that direction.

**
Yes, but we are not doing QM.
**

I was not talking about standard QM.

So, I have given some anwers, now it is up to you to come up at least a few concretos.

Careful

 

[Kea]

You forgot quite a few of the cosmological problems, and the only answer which had any content was your mention of MOND.

 

[Careful]

You forgot quite a few of the cosmological problems, and the only answer which had any content was your mention of MOND.

Euh excuse me?! I think the bounce at the classical singularity was a fairly nontrivial thing too - if some LQG thingie comes up with that, it is big news spread around everywhere! - or do you have problems with that ?

A few other questions of her:
(i) why do we live in 3+1 dimensions ? That does not even qualify as an urgent question in my view.
(ii) black holes : I ask you what *is* information in quantum gravity ? But if you consider guesses meaningful, then I would say information gets lost in the naive classical reasoning, but that ``quantum mechanically'' it could very well be that information is preserved for an asymptotic observer.
(iii) concerning the cosmological constant, as well 't Hooft and Adler persuasively argue that a deterministic quantum mechanics can tackle that one.

The rest of the questions are open in *cosmology* and it is presently very unclear whether something like a quantum gravity mechanism is needed for all of this (inflation could account for some of them). None of the QG approaches even make plausible suggestions in that direction. Actually, speculating about such things is pretty silly, there are more ``simple'' and basic questions to tackle first. So, I adressed 7 out of 10, some of the why questions remind me of ``why do we need God?''. One such question is why the gauge group of the standard model is SU(3)*SU(2)*U(1), interesting question but way too early to give a sensible answer to that.

Let's get back to the questions I posed you, since you were going to explain the physics of category theory.

Careful

 

[Kea]

I think the bounce at the classical singularity was a fairly nontrivial thing too...or do you have problems with that?

Yes, of course I do. It's not physical.

...you were going to explain the physics of category theory.

Actually, no, not right now. Believe it or not, I'm not here for your entertainment. Besides, it's quite clear that you've made no effort to understand this approach. I'm going to visit some barmaid friends. They like category theory.

 

[Careful]

**Yes, of course I do. It's not physical.**

Euuhhh why ? Do you actually know of this approach ? Could you shed your light upon this since I did not have much objections against it even though I think BM has some serious shortcomings (and did not take such line of thought seriously *further* when the authors went to the traditional ``multiparticle'' description - since the ``one-particle field'' contains them all).

***
Actually, no, not right now. Believe it or not, I'm not here for your entertainment. Besides, it's quite clear that you've made no effort to understand this approach. I'm going to visit some barmaid friends. They like category theory. **

This is unbelievable, I acknowledged from the beginning that I did not plunge into it for reasons which I made abundantly clear. Your were going to explain it, make it somehow plausibe : I am even prepared to forget my objections against the QM measurement and listen to your version of the Mach principle... Look, if I were genuinely not even slightly curious to know what the fuzz is all about, then I would not even have bothered reading about it and certainly I would not have made the effort to give my impressions. This is not an brainwrestling contest, I am genuinely giving you a chance to explain it - of course you will have to come up with good stuff (for a physicist) but that is only normal I think.

Careful

 

[Kea]

...then I would not even have bothered reading about it...

Careful, it's quite clear to everybody that you have not been reading about it. Reading one or two simple papers isn't going to bring you up to scratch on decades of interesting research.

 

[Careful]

Careful, it's quite clear to everybody that you have not been reading about it. Reading one or two simple papers isn't going to bring you up to scratch on decades of interesting research.

Sorry Kea, but your reaction is pretty poor which is also clear to everybody - I am not even contesting anything and it might very well be that category theory research is pretty interesting from a mathematical point of view. You did not adress any of my specific questions/points and merely keep on speaking about the interesting things happening - but actually refuse to give any specific example. You would better adress some specific points instead of telling everyone to read first tons of literature (sorry but this indicates poverty from your side, not vice versa).

And don't play with words : I clearly did read the Baez paper. If you simply cannot adress specific points then say so and the discussion is over.

Careful