Baratin and Freidel: a spin foam model of ordinary particle physics

[marcus]

I'll fetch some related links, in case something turns up or anyone is interested. I saw some recent papers by Speziale and also by Christensen and friends. UWO must be a good place to do computational quantum gravity, which ought to become important.

... why Simone Speziale and Dan Christensen are getting really good agreement with the graviton propagator based on calculations involving a single big 4-simplex, refinements of http://arxiv.org/abs/gr-qc/0508124" . They made a lot of progress on this last week: Dan's supercomputer calculations match what Simone is getting analytically....

Here are some Christensen links. He is at Uni Western Ontario---part in QG-physics and part in math+computer science. They have supercomputer facilities. Wade Cherrington is a grad student there, and Josh Willis from Ashtekar's Penn State institute is a post doc. If it turns out to be possible to numerically simulate the quantum evolution of a world geometry by means spin foam then I suppose this might eventually happen on a UWO cluster.
http://arxiv.org/abs/gr-qc/0512004
http://arxiv.org/abs/gr-qc/0509080
http://arxiv.org/abs/gr-qc/0508088
==========

Here are some Speziale links
1. gr-qc/0606074
A semiclassical tetrahedron
Carlo Rovelli, Simone Speziale
10 pages

2. gr-qc/0605123
Towards the graviton from spinfoams: higher order corrections in the 3d toy model
Etera R. Livine, Simone Speziale, Joshua L. Willis
24 pages, many figures

3. gr-qc/0604044
Graviton propagator in loop quantum gravity
Eugenio Bianchi, Leonardo Modesto, Carlo Rovelli, Simone Speziale
41 pages, 6 figures

4. gr-qc/0512102
Towards the graviton from spinfoams: the 3d toy model
Simone Speziale
8 pages, 2 figures
Journal-ref: JHEP 0605 (2006) 039

5. gr-qc/0508106
On the perturbative expansion of a quantum field theory around a topological sector
Authors: Carlo Rovelli, Simone Speziale
7 pages

6. gr-qc/0508007
From 3-geometry transition amplitudes to graviton states
Authors: Federico Mattei, Carlo Rovelli (CPT), Simone Speziale, Massimo Testa
18 pages
Journal-ref: Nucl.Phys. B739 (2006) 234-253

Here is another interesting thing that turned up:

... perhaps the best part is: Freidel now understands 2-Hilbert spaces and 2-groups, and he wants to keep studying models based on them!

Kea will be glad to hear that.
Don't let me get in the way if someone wants to be reply to the general sense of JB's post, I am just assembling some detail to think about in that connection.

 

[Sauron]

Just an small sugestion.

In my opinion the programing analogue of cathegory theory would be UML (uniform modelling language). It is fine to plot diagrams and clarify flow of information. But you can do everything just implementing the apropiate class.

Returngin to the maintopic, i have just made a first (and complete) reading of arXiv:gr-qc/0607014.

To say it easy, I had readed the talks in other thread about de-sitter but the by far the part which i understand less is the origin of the point lagrangian that they present in eq 3.1. I mean, it is basically the lagrangean of a classicla point particle carriying somehow information about so(3,1) álgebra or something similar? Wich sense makes that?

I see that they later show that it describes a particles with "all that good behaviours" but even so i don´t see clear that lagrangian (yeah, sure it is my fault).

And later, in chapter 5 when it makes a wilson loop with the exponential of that lagrangian, i simply don´t see the relation with the Feyman amplitudes. Maybe i need to read some previous papers? perhaps the ones aobut hidden quantum gravity in 3-d Feyman diagrams?

 

[Careful]

***
http://math.ucr.edu/home/baez/quantum_spacetime/" . I'm talking about how the category of Hilbert spaces (Hilb) and the category of n-dimensional cobordisms (nCob) are both monoidal categories with duals. The fact that Hilb has duals allows for quantum teleportation; the fact that nCob also has them is what allows you to straighten out a kink in a rope (in the case n = 1). ***

Yes, and as I said, I acknowledge that - I merely was commenting on how serious we should take these analogies (which I deduced for myself a few years ago while thinking about topology change).

***
This is simply a fact, not a "model" - and certainly not a model where quantum entangled particles are connected in some way, e.g. by an "invisible rope". Quantum entanglement arises from the fact that Hilb is non-cartesian, unlike the category of sets. nCob is also non-cartesian. **

Sure, but what is the spacetime interpretation you have in mind ?! I tried at least to offer some ways of looking at entanglement (or quantum teleportation) which would pave the road for such abstraction. I never claimed that your ``absolute truth'' was limited to those viewpoints, but said that it could be found by thinking in these ways (remember : abstraction of specifications of spin foam models). So, I have put myself in the weak position, not you; you have limited yourself to repeating the abstract results and their universality (as well as which of the latter pictures you do not have in mind) while I was interested in getting out the physics. I do not know what is worse, to keep on stressing the abstract results or the failure to recognize that the other party is begging for a specific way of looking at it (for example in terms of wormholes).

***
While these are just mathematical facts, they point the way towards models of quantum gravity, by showing us which class of mathematical structures combine the physically important features of general relativity and quantum mechanics. ***

I disagree here and I explained several times why - it is kind of silly you keep on repeating this without adressing these points - shows there was never an online discussion to begin with. I asked you how you would solve the problem of time while keeping measurement as it stands and still have a reasonable theory to end with (that is one which makes solid predictions). The reason why I ask you is that I came to the conclusion that doing so will require a profound change of the dynamics of measurement in QM (as some people in MIT try to realize). You could of course say : ``I limit myself to taking expectation values'' but then I would not see why one would not be pleased with merely imposing a sufficiently high UV cutoff (as well as a macroscopic nonlocality scale) in perturbative QG.

In the beginning you said you are not doing physics anymore, so why not tell what problems you see in it ??

Careful

 

[Kea]

I disagree here and I explained several times why - it is kind of silly you keep on repeating this without adressing these points - shows there was never an online discussion to begin with.

With you, there was never a discussion. Quite true.

I asked you how you would solve the problem of time while keeping measurement as it stands...

We are not keeping measurement as it stands, as I have said many times. Neither are we repeating in full detail here what has been said in many good papers, some of which I suggest you start reading.

The reason why I ask you is that I came to the conclusion that doing so will require a profound change of the dynamics of measurement in QM...

Correct.

 

[Careful]

**With you, there was never a discussion. Quite true.**

There has never been any discussion between category theorists and anyone here , since basically I am the only non-silent ``opposition''.

***
We are not keeping measurement as it stands, as I have said many times. Neither are we repeating in full detail here what has been said in many good papers, some of which I suggest you start reading.
****

I am not asking for any detail, I want a rough physical picture ! Explain in words what the mechanism is you have in mind for dealing with entanglement (and please do not refer me to the aforementioned papers, since these do not contain any such thing - there is only abstraction) and other crucial issues ...

**
Correct. **

So, please make a sketch of the mechanism *you* have in mind, then try to tell us why categories would come in handy here; not in the reverse order ! Roughly speaking, if you would not change measurement too drastically (no realism), then I guess you have to end up with something like an improved GRW, a scheme a la Penrose... (there are other possibilities too but those jump immediately to my mind). Anyway, something which would make the total dynamics non-linear.

I think it would be good for many people here if you would start by doing that : put some of your cards on the table - let us talk about physics first and suggest then how your categories come out in a *deeper* way, just as this happened in the de Rahm theorem . Referring to papers is not the way to go, especially when no response to criticism comes out.


Careful

 

[marcus]

Probably a bunch of us have had a look at Baez paper QUANTUM QUANDARIES, which introduces the notion of a *-category.
nCob and Hilb are both categories of this sort.

today Robert Coecke posted a paper developing similar themes.
It uses different terminology and a somewhat more restrictive definition.

the notion of a "dagger-compact" category

I still have to find how to type a dagger. [tries various things]
It looks like it is OPTION TEE!

OK Coecke, I mean Okey Dokey. this paper will probably turn out to be used and cited some in the process of categories permeating physics through something Coecke calls CATEGORICAL SEMANTICS.

So I had better post the abstract.

BTW Coecke's reference [4] is Baez Quantum Quandaries.
and his reference [30] is THE DISENCHANTMENT OF JOHN VON NEUMANN WITH HILBERT SPACES.
von Neumann became dubious of Hilbert spaces and declared they were not where it's at.
(that is, not where Quantum Mechanics is at)

 

[marcus]

Here is the Coecke et al new paper:
http://arxiv.org/abs/quant-ph/0608035
Quantum measurements without sums
Bob Coecke, Dusko Pavlovic
36 pages and 46 pictures; earlier version circulated since November 2005 with as title 'Quantum Measurements as Coalgebras''. Invited paper to appear in: The Mathematics of Quantum Computation and Technology; Chen, Kauffman and Lomonaco (eds.); Taylor and Francis

---sample exerpt from page 2 of the article---
Ever since John von Neumann denounced, back in 1935 [30], his own foundation of quantum mechanics[/color] in terms of Hilbert spaces, there has been an ongoing search for a high-level, fully abstract formalism of quantum mechanics. With the emergence of quantum information technology, this quest became more important than ever. The low-level matrix manipulations in quantum informatics are akin to machine programming with bit strings from the early days of computing, which are of course inadequate. 1

...
...
A recent research thread, initiated by Abramsky and the first author [2], aims at recasting the quantum mechanical formalism in categorical terms. The upshot of categorical semantics is that it displays concepts in a compositional and typed framework. In the case of quantum mechanics, it uncovers the quantum information-flows [6] which are hidden in the usual formalism. Moreover, while the investigations of quantum structures have so far been predominantly academic, categorical semantics open an alley towards a practical, low-overhead tool for the design and analysis of quantum informatic protocols, versatile enough to capture both quantitative and qualitative aspects of quantum information [2, 7, 10, 13, 31]. In fact, some otherwise complicated quantum informatic protocols become trivial exercises in this framework [8]. On the other hand, compared with the order-theoretic framework for quantum mechanics in terms of Birkhoff-von Neumann’s quantum logic [29], this categorical setting comes with logical derivations, topologically embodied into something as simple as “yanking a rope” 2. Moreover, in terms of deductive machanism, it turns out to be some kind of “super-logic” as compared to the Birkhoff-von Neumann “non-logic”.
---endquote---

Baez was talking about stretching out a piece of wet spaghetti. curious propositions in quantum theory, seeming paradoxes, become trivial exercises as Coecke says. Baez was trying to get that idea across---basically one of the reasons why one might see categorical semantics filter into physics.

Reference [30] in the above exerpt is:
"[30] Rédei, M. (1997) Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead). Studies in History and Philosophy of Modern Physics 27, 493–510. "

Here is the abstract:
"Sums play a prominent role in the formalisms of quantum mechanics, be it for mixing and superposing states, or for composing state spaces. Surprisingly, a conceptual analysis of quantum measurement seems to suggest that quantum mechanics can be done without direct sums, expressed entirely in terms of the tensor product. The corresponding axioms define classical spaces as objects that allow copying and deleting data. Indeed, the information exchange between the quantum and the classical worlds is essentially determined by their distinct capabilities to copy and delete data. The sums turn out to be an implicit implementation of this capabilities. Realizing it through explicit axioms not only dispenses with the unnecessary structural baggage, but also allows a simple and intuitive graphical calculus. In category-theoretic terms, classical data types are dagger-compact Frobenius algebras, and quantum spectra underlying quantum measurements are Eilenberg-Moore coalgebras induced by these Frobenius algebras."

 

[marcus]

the folklore (and I think this has been reliably confirmed by at least one scholar) is that in 1925 around the time he devised matrix mechanics version of QG Heisenberg did not know what a matrix was
and had never heard of Hilbert spaces.

According to the JB history draft, page 5, Heisenberg came to show a formula to Max Born* who informed him that he had "re-invented matrix multiplication".

Apparently the young physicists inventing QM at that time hadn't heard of Hilbert spaces. It was John von Neumann, a mathematician, who introduced them and showed them how to formulate QM with operators on a Hilbert space. However soon afterwards, von Neumann became disenchanted with the Hilbertspace formulation and wanted things to be done differently. BUT BY THEN IT WAS TOO LATE.
The whole pack was already off like hounds after a fox.

please correct any historical errors.*Max Born was Heisenberg's mentor at Göttingen, where he was a visiting student and later got a job. Heisenberg's thesis advisor was Sommerfeld, in Munich, and his thesis was in hydrodynamics. After it was accepted in 1923. he immediately returned to Göttingen and worked as Born's assistant..
http://www.aip.org/history/heisenberg/p06.htm
It seems that although Max Born served as a mentor to the young Heisenberg, he did not supervise his PhD thesis.
http://nobelprize.org/nobel_prizes/physics/laureates/1954/born-bio.html

=============

at first glance, it looks to me like what JB was calling a "star-category", Robert Coecke would prefer to call a "dagger-category".
I think someone with an ear for English will be apt to prefer "star-category" to "dagger-category" for several reasons. Tthe phrase rings better---with a better assortment of vowells. It has fewer syllables. The concept is all about things like adjoint of an operator A, something often written A*, the complex conjugate transpose of a matrix. Mathematicians frequently use the asterisk * for duals and adjoints and such.
So if Coecke insists on the nomenclature "compact" then a sensible compromise would be "compact star category"------instead of "dagger-compact category"----but we will just have to wait and see

here is a picture of Bob Coecke (oxford computing lab)
http://web.comlab.ox.ac.uk/oucl/people/bob.coecke.html
he has an impressive list of publications since around 1999
http://arxiv.org/find/grp_physics/1/au:+Coecke_Bob/0/1/0/all/0/1

the co-author Dusko Pavlovic is at Kestrel Institute in Palo Alto. It is the not-for-profit institute connected with the software development company Kestrel Development. Both wings of Kestrel sound like interesting places to work.

 

[john baez]

star versus dagger, compact versus closed

at first glance, it looks to me like what JB was calling a "star-category", Robert Coecke would prefer to call a "dagger-category"

I think so. A star-category is a category where any morphism

f: x -> y

can be "run in reverse" to give a morphism

f*: y -> x

and we have

f** = f

(fg)* = g*f*

Is this the same as Coecke's "dagger-category"?

I think someone with an ear for English will be apt to prefer "star-category" to "dagger-category" for several reasons. The phrase rings better---with a better assortment of vowels. It has fewer syllables.

That's true - it's also been in use longer! With no offense intended, I think people working on categories and quantum computation are reinventing certain concepts developed by people working on http://arxiv.org/abs/math.CT/9812040" . This is good: it means these concepts are really important. But, it causes some notational conflicts.

The concept is all about things like adjoint of an operator A, something often written A*, the complex conjugate transpose of a matrix.

Exactly, that's the main example - but physicists often write this with a dagger instead of a star.

So if Coecke insists on the nomenclature "compact" then a sensible compromise would be "compact star category"------instead of "dagger-compact category"----but we will just have to wait and see.

Compact categories have been around for a long, long time - they're categories where objects have nice duals. How did the term "compact" get used for this? Well, for one thing, compact categories are a special case of closed categories, where given two objects x,y you have an object

HOM(x,y)

that acts like "the maps from x to y".

For example, consider the category of vector spaces. Given two vector spaces x and y, HOM(x,y) is the vector space of linear operators from x to y.

A closed category is called "closed" because normally we have a set of morphisms from x to y, but now we have an object of morphisms from x to y, so we don't have to leave our category to talk about "hom"! So, a closed category is like its own self-contained universe! Cool, huh?

Of course the classic example of a closed category is the category of sets, where there's a set of functions from a set x to a set y. When they invented closed categories back in 1966, http://citeseer.ist.psu.edu/context/20076/0" were trying to let other categories be "self-contained" like this. It's one step towards dethroning the category of sets - getting it to stop acting better than everyone else.

Every topos is a closed category... that was a later step towards dethroning the category of sets.

A compact category is a special sort of closed category where

HOM(x,y) = x* tensor y

For example, this is true for the category of finite-dimensional vector spaces. It's not true for the category of sets, nor for most topoi. Topoi are cartesian closed, not compact, so they embody intuitiionistic logic, not quantum logic.

Now I can finally explain the term "compact" - this is taking long than I expected.

Since "compact" sets in a topological space are specially nice "closed" sets, when people discovered specially nice closed categories they decided to call them compact!

In other words, it's just an erudite joke, of the sort nobody finds funny except mathematicians.

By the way, I haven't given the actual precise definitions of closed and compact categories here, just the intuitions. I did give the precise definition of a star-category, though.

Personally, I often call a star-category a "category with duals for morphisms", and a compact category a "category with duals for objects".

In quantum mechanics we often want categories with both, which I call "categories with duals". I think Abramsky and Coecke call these "dagger-compact categories", or maybe "strongly compact categories". The terminology is more confusing than the actual ideas.

 

[Kea]

Coecke seems to use daggers (\dagger) on morphisms (which one gets from a functor C^{\textrm{op}} \rightarrow C for a symmetric tensor category) and stars (*) on objects as part of the compact structure. One then has

f^{\dagger} = (f^{*})_{*} = (f_{*})^{*}

which is explained on pages 7 and 8. Most physicists should prefer this to the reverse, so I think he's made a great effort to sort out the notational headaches.

The funny trapezium shapes are very clever, because then little diamond pieces can form and these represent scalars which can float about just like loops in tangle diagrams! Take a look at page 29.

 

[john baez]

Is this a dagger which I see before me?

The title of my post is quote from Shakespeare's Macbeth. Clearly Macbeth was reading Coecke's paper at the time, puzzled by why Coecke failed to use star-categories.

I still have to find how to type a dagger. [tries various things]
It looks like it is OPTION TEE!

You can also use TeX, with the command \dagger:

\dagger

This paper will probably turn out to be used and cited some in the process of categories permeating physics through something Coecke calls CATEGORICAL SEMANTICS.So I had better post the abstract.

Thanks! I'm not sure this thread on Baratin-Freidel is the best place, since it'll take quite a while for categorical semantics, quantum computation, spin foams and MacDowell-Mansouri gravity to blend into one grand subject... if they ever do! But heck, let's be optimists - especially since it's Saturday night here in Shanghai, and beer is cheap: about 40 cents for a 32-ounce bottle.

After Jeff Morton and Derek Wise, my next grad student in line to finish up is Mike Stay. It'll take him a few more years. I'm working with him on quantum computation and categorical semantics.

Categorical semantics is where you describe a theory by giving a category C with some extra structure, and looking at functors that preserve this structure:

F: C -> D

where D is another category of the same sort of structure. You think of C as a theory and F as a model of this theory in D. Here "theory" and "model" are being used the way logicians use these terms, not physicists!

Lawvere invented this idea in his radical http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html" - a setup for describing very general sorts of algebraic gadgets and proving theorems about all these kinds of gadgets in one fell swoop.

There's been a lot of work on categorical semantics since then, especially by computer scientists.

More recently physicists have gotten interested in this stuff, for example when C and D are "symmetric monoidal categories with duals" and F was "symmetric monoidal functor preserving duals". If you take

C = nCob, D = Hilb

then a functor F of this sort is called a topological quantum field theory.

(I explained this in an amusing tale involving a wizard and his apprentices http://math.ucr.edu/home/baez/qg-winter2001/qg11.1.html" .)

Note that physicists use the term "topological quantum field theory" for what the logicians would call a "model of the theory nCob in Hilb".

Anyway, since I've been emphasizing this relationship between categorical semantics and physics, computer scientists have started inviting me to their conferences, which is very nice. I spent a month earlier this year in Marseille for that very reason, and I gave a course on http://math.ucr.edu/home/baez/universal/" in which I explained this stuff in much, much more detail.

Is this a dagger which I see before me,
The handle toward my hand? Come, let me clutch thee.
I have thee not, and yet I see thee still.
Art thou not, fatal vision, sensible
To feeling as to sight? or art thou but
A dagger of the mind, a false creation,
Proceeding from the heat-oppressed brain?
I see thee yet, in form as palpable
As this which now I draw: \dagger


- William Shakespeare
​

 

[marcus]

this thread is partly about the expected 4D paper of Aristide Baratin and Laurent Freidel

and also has come to be about the percolation of categorics into physics

a propos which, there was TWF 236 and also some entertaining anecdotal stuff in a SPR FOLLOWUP to TWF 236----post #5 on the thread---which was so funny I want to quote an exerpt:

===quote Baez TWF 236 followup===
>(I don't recall if it was Rutherford or Lord Kelvin who
>claimed this.)

Lord Kelvin is mainly noted for having dismissed *vectors* as
unnecessary to physics. He wrote:
Quaternions came from Hamilton after his really good work had been
done; and though beautifully ingenious, have been an unmixed evil
to those who have touched them in any way, including Maxwell.
Vector is a useless survival, or offshoot from quaternions, and has
never been of the slightest use to any creature.

To understand this, remember that J. Willard Gibbs, the first person
to get a math PhD in the USA, introduced the modern approach to vectors
around 1881, long after Hamilton's quaternions first became popular. He
took the quaternion and chopped it into its "scalar" and "vector" parts.

Vectors are another great example of a convenience that's so convenient
that they're now seen as a necessity.

It's mainly the American physicist John Slater, inventor of the "Slater
determinant", who is famous for having dismissed groups as unnecessary to physics. He wrote:

It was at this point that Wigner, Hund, Heitler, and Weyl entered the picture with their "Gruppenpest": the pest of the group theory [actually, the correct translation is "the group plague"] ... The authors of the "Gruppenpest" wrote papers which were incomprehensible to those like me who had not studied group theory... The practical consequences appeared to be negligible, but everyone felt that to be in the mainstream one had to learn about it. I had what I can only describe as a feeling of outrage at the turn which the subject had taken ... it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the problem. As I heard later, there were remarks made such as "Slater has slain the 'Gruppenpest'". I believe that no other piece of work I have done was so universally popular.

And now, of course, it's categories that some physicists dismiss, just
as they're catching on.

So, judging by the history, you can be almost sure that if a bunch of physicists
angrily dismiss a branch of mathematics as useless to physics,
it's useful for physics.
The branches of math that don't yet have applications to physics don't arouse such controversy!
===endquote===

I highlighted some memorable parts. Especially nice about "vector is a useless survival...never been the slightest use to any creature."

 

[Sauron]

I see interesting these historical apportations. But I think that preciselly history, recent history, is the reason behind these reluctance to quaternions.

We have had that from the beginning of the century to the eighties in physics there has been minor changes inthe math background needed by a physician.

Only group theory, in a very elementary framework and diferential geometry if you are into RG were real innovations (hilbert spaces in most places are not much more that a convenient framework ofr the join of fourireer analisis and linear álgebra).

And with these very modest bagage it was created relativity, quantum mechanics, modern statisticla mechanics and quantum field theory. That is, the whole amount of experimentally tested physics.

Them, in the eigties it came a modernization of the math background of the theoretical physicans pusing them into the realm of a math usually only teached in postgraduat of math courses (of course there was a few pioners as hawking and penrose, or maybe people working on monopoles ansuch that).

And with all that pletora of maths there has not beenany mayor advance on tested theoretical physic.

I suspect people is just tired of learning just more maths and want phyisical intuition, development of ideas with current background in math (wich is alerady a lot of math) and some experimental result.

On the other side theoretical physics has gone mainly with very formal math (I really like topology, but "funcional analisys" kind of things are a bit annoying).

And on the other side we have a lot of new math which has had a marginal impact intheretical physic. I refere to "chaos" math which is lot less formal and has had a lot more of impact inthe way of think about natural phenomena.

I currently am teaching ecologist docotorates some of these new maths (dynamical systems, linear programing, markov chains,game theory, graph theory, even "complexity", etc,of course in a light way) and I find it mucho more atractive to be learned than category theory. And I also find that with time these math will play role in quantum gravity.

But well, if it becomes clear tahat we need to learn in deep categories we will do. I really thank the efforts of Jonh Baez by beeing so patient and pedagogical. I guess he hast just had a bad luck with the time to introduce a new math language to the physical comunity.

 

[marcus]

Coecke seems to use daggers (\dagger) on morphisms (which one gets from a functor C^{\textrm{op}} \rightarrow C for a symmetric tensor category) and stars (*) on objects as part of the compact structure. One then has

f^{\dagger} = (f^{*})_{*} = (f_{*})^{*}

which is explained on pages 7 and 8. Most physicists should prefer this to the reverse, so I think he's made a great effort to sort out the notational headaches.
...

Thanks Kea,
perhaps it is all for the best. (in any case what can one do?)
Two more Coecke papers appeared today on arxiv. Here is one of them, of possible interest here.
http://arxiv.org/abs/quant-ph/0608072
POVMs and Naimark's theorem without sums
Bob Coecke, Eric Oliver Paquette
"We introduce POVMs within the purely graphical categorical quantum mechanical formalism in terms of dagger-compact categories (cf. quant-ph/0402130, quant-ph/0510032 & quant-ph/0608035). Our definition is justified by two facts: i. We provide a counterpart to Naimark's theorem, which establishes a bijective correspondence between POVMs and abstract projective measurements on an extended system; ii. In the category of Hilbert spaces and linear maps our definition coincides with the usual one."

in case it is wanted, here is the Wiki article on Positive Operator-Valued Measures
http://en.wikipedia.org/wiki/POVM
POVM is an important concept in quantum theory, basic to formalizing quantum measurement.
I interpret his targeting this for the "without sums" treatment as his aiming to blitz QM formalism
and hit several major landmarks in quick succession.

Again this paper cited JB's Quantum Quandaries, as its reference [4]

I don't know that there is any special reason to discuss the other Coecke paper that came on arxiv today
http://arxiv.org/abs/math.LO/0608166
the abstract refers to the muddy children problem. If that is familiar you might want to look it up.

 

[bananan]

when's that paper being published?

Thanks Kea,
perhaps it is all for the best. (in any case what can one do?)
Two more Coecke papers appeared today on arxiv. Here is one of them, of possible interest here.
http://arxiv.org/abs/quant-ph/0608072
POVMs and Naimark's theorem without sums
Bob Coecke, Eric Oliver Paquette
"We introduce POVMs within the purely graphical categorical quantum mechanical formalism in terms of dagger-compact categories (cf. quant-ph/0402130, quant-ph/0510032 & quant-ph/0608035). Our definition is justified by two facts: i. We provide a counterpart to Naimark's theorem, which establishes a bijective correspondence between POVMs and abstract projective measurements on an extended system; ii. In the category of Hilbert spaces and linear maps our definition coincides with the usual one."

in case it is wanted, here is the Wiki article on Positive Operator-Valued Measures
http://en.wikipedia.org/wiki/POVM
POVM is an important concept in quantum theory, basic to formalizing quantum measurement.
I interpret his targeting this for the "without sums" treatment as his aiming to blitz QM formalism
and hit several major landmarks in quick succession.

Again this paper cited JB's Quantum Quandaries, as its reference [4]

I don't know that there is any special reason to discuss the other Coecke paper that came on arxiv today
http://arxiv.org/abs/math.LO/0608166
the abstract refers to the muddy children problem. If that is familiar you might want to look it up.

 

[marcus]

when's that paper being published?

Obviously they all have been published in PDF on the web, so I think you must mean published in paper, and I don't know. but tell me which particular paper you want to know about and I will see if I can figure out.

 

[bananan]

Obviously they all have been published in PDF on the web, so I think you must mean published in paper, and I don't know. but tell me which particular paper you want to know about and I will see if I can figure out.

With any luck, sometime soon you can read this paper on the arXiv:

Aristide Baratin and Laurent Freidel
Hidden quantum gravity in 4d Feynman diagrams: emergence of spin foams

The idea is that any ordinary quantum field theory in 4d Minkowski spacetime can be reformulated as a spin foam model. This spin foam model is thus a candidate for the G -> 0 limit of any spin foam model of quantum gravity and matter!

In other words, we now have a precise target to shoot at. We don't know a spin foam model that gives gravity in 4 dimensions, but now we know one that gives the G -> 0 limit of gravity: i.e., ordinary quantum field theory. So, we should make up a spin foam model that reduces to Baratin and Freidel's when G -> 0.

 

[marcus]

when's that paper being published?

I see what you are talking about now. Your originally quoted my post #114 which mentioned Coecke papers so I thought you were talking about those.
You are asking about the Baratin Freidel paper that John Baez mentioned at the start of this thread!

I am sorry to say but I do not know and I cannot think of any way to get a good estimate.
My guess is EARLY OCTOBER (this is based on nothing else than my past experience of the spacing of major Freidel papers.)

 

[marcus]

I see what you are talking about now. Your originally quoted my post #114 which mentioned Coecke papers so I thought you were talking about those.
You are asking about the Baratin Freidel paper that John Baez mentioned at the start of this thread!

I am sorry to say but I do not know and I cannot think of any way to get a good estimate.
My guess is EARLY OCTOBER (this is based on nothing else than my past experience of the spacing of major Freidel papers.)

Back some time ago (August), we were discussing the 4D Baratin Freidel paper, which John Baez called our attention to.

I know the paper has been circulated in draft, at least to a few people. Getting comments, I guess---maybe suggestions for changes and additions.

I am expecting "early October" for the appearance of this paper.

but one can't know. just hope no hitches developed. I'm actually curious and a bit impatient. Hope all is well with them and with the research, and that we will soon see the results!

 

[marcus]

Amazingly, John Baez started this thread 16 June introducing us to the ideas of the Baratin-Freidel paper. and that was almost 5 months before the paper actually appeared!

It is called being forehanded.
When you say thank you to JB the feeling is always a little like to the Lone Ranger, he already went somewhere (like down to the n-cat saloon) and left a silver bullet on the mouse-pad

 

[marcus]

Baez started the thread by talking about this paper, which has just come out (and maybe the follow-up paper which he may be co-authoring with B and F)

He called the thread
Baratin and Freidel: a spin foam model of ordinary particle physics

Now that the paper is out, perhaps we should ask some questions like

What does the paper do that we expected?
Does it do anything that was not anticipated in our (or JB's) discussion?
Does it leave anything out, that we expected----maybe putting it off to a later paper?

Is this really a (background independent) model of ordinary particle physics?[/color]

If it is, that is a first----ordinary particle physics is built on rigid pre-ordained normally flat background geometry----but the spinfoam approach is manifestly a background independent formulation. It does not appeal at any point to a set-up geometry.

It looks like Freidel COAXED A SENSE OF AMBIENT GEOMETRY INTO THE FEYNMAN DIAGRAM ITSELF. So that the Feynman diagram knows enough about the geometry that it doesn't have to have it spelled out ahead of time. And it has a ZERO-GRAVITY LIMIT where you let the Newton parameter G_N go to zero, so space flattens out, and you get the same amplitudes as you would in usual QFT on flat spacetime.

What seems to be missing, for me, in the paper is a confirmation that the spinfoam model has deformed Poincare symmetry. In the earlier "Hidden" paper, where B and F dealt with the 3D case, there was the expectation of an energy-dependent speed of light----something that GLAST could test.
I don't see that here. So I am asking anyone else who has been reading the paper what they think.
==============

for reference, here is the new paper:
http://arxiv.org/abs/hep-th/0611042
Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams
Aristide Baratin, Laurent Freidel
28 pages, 7 figures

"We show how Feynman amplitudes of standard QFT on flat and homogeneous space can naturally be recast as the evaluation of observables for a specific spin foam model, which provides dynamics for the background geometry. We identify the symmetries of this Feynman graph spin foam model and give the gauge-fixing prescriptions. We also show that the gauge-fixed partition function is invariant under Pachner moves of the triangulation, and thus defines an invariant of four-dimensional manifolds. Finally, we investigate the algebraic structure of the model, and discuss its relation with a quantization of 4d gravity in the limit where the Newton constant goes to zero."

Here is the promised follow-up, their reference [23], expected to be co-authored with JB
[23] J. Baez, A. Baratin, L. Freidel, On the representation theory of the Poincaré 2-group, To appear.

Here is the earlier "Hidden" paper by B and F, dealing with the 3D case
http://arxiv.org/abs/gr-qc/0604016
Hidden Quantum Gravity in 3d Feynman diagrams
Aristide Baratin, Laurent Freidel
35 pages, 4 figures

"In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincaré group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincaré group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results."

I may be mistaken---maybe I should not have been expecting news about an energy-dependent speed of light. I don't see any mention of that in the abstract, have to check.

 

[marcus]

...
Here is the promised follow-up, their reference [23], expected to be co-authored with JB

[23] J. Baez, A. Baratin, L. Freidel, On the representation theory of the Poincaré 2-group, To appear.

...

In his most recent, This Week's Finds #269, John Baez remarks that he is hard at work with Baratin, Freidel, and Wise on a paper on representations of 2-groups.

It's as if there has been a hiatus of about two years! 2006 to 2008. Good luck to them! Maybe some of the stuff in this thread will provide a useful review and warmup for anyone who wants to get prepared to understand the 2-group paper they are working on, when it is ready. In the brief mention in TWF #269 the word "gnarly" is used, so be forewarned.

 

[MTd2]

In his most recent, This Week's Finds #269, John Baez remarks that he is hard at work with Baratin, Freidel, and Wise on a paper on representations of 2-groups.

It's as if there has been a hiatus of about two years! 2006 to 2008. Good luck to them!

You know that doesn't mean he is back to research on LQG neiter Spin Foam stuff.

 

[marcus]

Certainly MTd2! You make an obvious point. The representation of 2-groups is a math research topic of general interest. It is not an applied research topic and the interest is obviously not limited to Quantum Gravity. Let me say things a little more clearly, to make sure you do not misunderstand me:
In 2006, Baez, Baratin and Freidel had a paper in preparation on the representation of 2-groups. This paper did not appear. There was a hiatus. Now it seems that they are again working on the representation of 2-groups---and Derek Wise has joined them making a fourth author.

You, in effect, raise the issue of what relation exists between the general math problem of 2-groups on the one hand, and quantum gravity on the other. That's a very interesting question that is talked a bit about in this thread. I think I understand the relation a little, though not completely. This thread is, in a sense, about the quantum gravity motivation for exploring 2-groups.

One way to understand the QG motivation, if you don't want to read the beginning of the thread, where JB explains it, is to look at the research history of the people. The research interests of Freidel, Baratin, and Wise are primarily in QG----Freidel especially in spinfoams.
In 2005 Freidel (with Livine and others) got a very interesting result including matter in spinfoam at 3D. Roughly speaking he found that in 3D the spinfoams of gravity and the Feynman diagrams of matter are, at a basic level, the same entity. A spinfoam is a combinatorial structure colored with group representations. Baratin was a postdoc working with Freidel part of that time.

The problem, which Freidel, Baratin, and others faced at that time was how to extend the results to 4D. One idea of how to do this involved coloring not only with group representations but with 2-group representations.

But there is basic mathematical ground-work to be done on 2-group representation theory before that application can be tackled. One probably needs to be able to classify the representations of the Poincaré 2-group. At least. In order to make it work. Not being an expert, I cannot tell, but I think that as the solution of an important pure math problem this would be noteworthy----in part because of the quantum gravity motivation but also for general aesthetic reasons. However the problem may not be tractable! It may be hellishly difficult!

Back in 2005 Derek Wise, another QG researcher, was doing his PhD thesis with John Baez, working on a different quantum gravity problem (not spinfoam) but one which I think could also use 2-groups. His thesis-related work (2007) is interesting, you might enjoy looking it up. Now postdoc at UC Davis with Steve Carlip.

So now we have a hint (just a brief mention) that JB might be working (he used the words "hard" and "gnarly") with these 3 quantum gravitists on a basic mathematical problem with strong QG motivation. What will come of this? Will they succeed? Will the work reach a satsifactory conclusion? Whatever they are working on seems to be tough, because the paper cited as in prep, back in 2006, never appeared. However Laurent Freidel is very stubborn. (I have watched his research since 2003 and can tell at least that much about him.) As bystanders, you and I are allowed to make bets, MTd2. I will bet that this time the paper will take shape to the authors' satisfaction, and we will see it in the next 6-12 months. Possibly earlier. So now would be the time to refresh one's ideas about 2-groups and about Freidel's way of uniting spinfoams with Feynman diagrams and establishing ordinary QFT in the spinfoam context.

 

[MTd2]

uniting spinfoams with Feynman diagrams and establishing ordinary QFT in the spinfoam context.

So that, you could also study supergravity from the point of view of spianfoams?

"A spinfoam is a combinatorial structure colored with group representations."

I recall reading the names "colored" and "diagrams" close to each other, in a method to help calculate the away divergencies in supergravities N=8, in one of those articles from Carrasco. I don't understand much, and I know it is not related at all to spinfoams. But what you said rhymes with that.

 

[marcus]

So that, you could also study supergravity from the point of view of spinfoams?
...

I don't have a lot of useful information about this. Supergravity has already been studied in the LQG context. I don't happen to have the links but there are papers and I guess any of us could dig them up with a keyword search. As a general framework LQG is compatible with both SUGRA and with extra dimensions. So presumably spinfoam would be equally.

But as far as I know, those papers go way back. At some point, probably in the 1990s or anyway before 2003, somebody checked to make sure LQG could accommodate SUGRA and D>4. But I don't know of any recent interest in that.

Here is a possible suggestion, where you might find something: check out the September 2008 Sussex workshop. I'll get a link. It includes top leaders in quantum gravity research like John Barrett, Renate Loll, and Laurent Freidel. Since N=8 SUGRA has been in the news a lot, if it holds any promise for non-perturbative QG then I would guess it would come up in the Sussex workshop. I've seen signs that N=8 supergravity is on the agenda, so let's keep an eye out for it.

If anything good can come out of cross-fertilization between different lines of nonperturbative field theory research, this Sussex workshop is going to exemplify it and set a pattern for the future. So check out the schedule (I expect you already have, actually!) Here's a post of mine with links:

the next major workshop/conference that I know about, is the one in Sussex 17-19 September
I posted an announcement about this in the ANNOUNCEMENTS thread back in June, month before last.
What I want to do here is study the topics, focus, and lineup of speakers for clues about where the field is going
Continuum and Lattice Approaches to Quantum Gravity
http://www.ippp.dur.ac.uk/Workshops/08/CLAQG
Among other things it will feature talks by
* Jan Ambjorn (NBI Copenhagen)
* John Barrett (U Nottingham)
* Laurent Freidel (ENS Lyon and Perimeter Institute)
* Renate Loll (U Utrecht)
* Max Niedermaier (U Tours)
* Roberto Percacci (SISSA Trieste)
* Martin Reuter (U Mainz)
* Thomas Thiemann (AEI Golm and Perimeter Institute)

You can see the emphasis
Triangulations----Ambjorn, Loll
Asymptotic Safety----Reuter, Percacci, Niedermeyer,
Spinfoam---Freidel, Barrett
canonical LQG---Thiemann

The three days of talks will be preceded by a school 15-16 September, to provide extra preparation for participants who wish it
Non-perturbative Methods in Quantum Field Theory
http://www.ippp.dur.ac.uk/Workshops/08/NPMQFT
Some of the lectures will be as follows:

* Basics of the non-perturbative renormalisation group (D. Litim, U Sussex)
* Basics of the Renormalization Group for QCD and confinement (J.M. Pawlowski, U Heidelberg)
* Basics of QCD on the lattice (O. Philipsen, U Muenster)
* Basics of asymptotic safety for gravity (M. Niedermaier, U Tours)
* Basics of the Renormalization Group for quantum gravity (M. Reuter, U Mainz)
* Basics of lattice quantum gravity I (R. Loll, U Utrecht)
* Basics of lattice quantum gravity II (J. Barrett, U Nottingham)

...

Here is the program:
http://www.ippp.dur.ac.uk/Workshops/08/CLAQG/Programme/

I see that Bjerrum-Bohr, from Princeton, is on the program. Let me see what his line of research is.

 

[marcus]

Ahah! I thought I remembered that Bjerrum-Bohr was into N=8 SUGRA!

So that too is part of the Sussex workshop. Look at Bjerrum-Bohr's recent papers:

1. arXiv:0806.1726 [ps, pdf, other]
Title: On Cancellations of Ultraviolet Divergences in Supergravity Amplitudes
Authors: N. E. J. Bjerrum-Bohr, Pierre Vanhove
Comments: Latex. 12 pages, 1 figure. Contribution to the proceedings of the 3rd meeting of the RTN `` Constituents, Fundamental Forces and Symmetries of the Universe'' in Valencia (Spain) and Quarks 2008 at Sergiev Posad (Russia). v2: minor corrections
Subjects: High Energy Physics - Theory (hep-th)
2. arXiv:0805.3682 [ps, pdf, other]
Title: Absence of Triangles in Maximal Supergravity Amplitudes
Authors: N. E. J. Bjerrum-Bohr, Pierre Vanhove
Comments: 16 pages, RevTeX4 format
Subjects: High Energy Physics - Theory (hep-th)
3. arXiv:0802.0868 [ps, pdf, other]
Title: Explicit Cancellation of Triangles in One-loop Gravity Amplitudes
Authors: N. E. J. Bjerrum-Bohr, Pierre Vanhove
Comments: 25 pages. 2 eps pictures, harvmac format. v2: version to appear in JHEP. Equations (3.9), (3.12) and minor typos corrected
Subjects: High Energy Physics - Theory (hep-th)
4. arXiv:0709.2086 [ps, pdf, other]
Title: Analytic Structure of Three-Mass Triangle Coefficients
Authors: N. E. J. Bjerrum-Bohr, David C. Dunbar, Warren B. Perkins
Comments: 22 pages; v3: NMHV n=point expression added. 7 point expression removed
Subjects: High Energy Physics - Phenomenology (hep-ph)
5. arXiv:gr-qc/0610096 [ps, pdf, other]
Title: On the parameterization dependence of the energy momentum tensor and the metric
Authors: N. E. J. Bjerrum-Bohr, John F. Donoghue, Barry R. Holstein
Comments: 8 pages, 2 figures
Subjects: General Relativity and Quantum Cosmology (gr-qc)
6. arXiv:hep-th/0610043 [ps, pdf, other]
Title: The No-Triangle Hypothesis for N=8 Supergravity
Authors: N. E. J. Bjerrum-Bohr, David C. Dunbar, Harald Ita, Warren B. Perkins, Kasper Risager
Comments: 43pages
Journal-ref: JHEP 0612 (2006) 072
Subjects: High Energy Physics - Theory (hep-th)

 

[MTd2]

Marcus, I guess this is very pertinent:

"Strings, quantum gravity and non-commutative geometry on the lattice
Authors: J. Ambjorn
(Submitted on 9 Jan 2002)

Abstract: I review recent progress in understanding non-perturbative aspects of string theory, quantum gravity and non-commutative geometry using lattice methods."

http://arxiv.org/PS_cache/hep-lat/pdf/0201/0201012v1.pdf

 

[marcus]

I think what we need to do, to keep this thread on topic, is to see how things relate what what Baez calls higher gauge theory---or with 2-groups.
The most basic way to look at the topic is it has to do with labeling spinfoams with reps of 2-groups instead of ordinary groups.
And doing gauge theory with 2-groups instead of ordinary groups, which I guess for brevity sake you could call 1-groups.

As I see it, it is still undecided whether quantum gravity NEEDS 2-groups. Maybe they are the key to success, maybe not. In any case they represent potentially powerful new mathematics and they will be developed (by people with good mathematical instincts) and they will be useful for something----maybe understanding space and matter fields, maybe something else.

I don't want to spend all my time thinking about 2-groups, but on the other hand I want to stay alert and interested in case any news comes up. that is the reason for keeping tabs on this thread.

 

[MTd2]

I think what we need to do, to keep this thread on topic

So, why you posted about a conference on "Continuum and Lattice Approaches to Quantum Gravity"?

 

[marcus]

Look back at your post #125, MTd2
I posted about that conference because I was trying to respond to this post of yours, which was off topic.

So that, you could also study supergravity from the point of view of spianfoams?

"A spinfoam is a combinatorial structure colored with group representations."

I recall reading the names "colored" and "diagrams" close to each other, in a method to help calculate the away divergencies in supergravities N=8, ...

I realized later I should not have tried to respond to a question about supergravity, because it is unrelated to the thread. But at that time I wanted to try to respond, so I mentioned this conference that is a kind of bridge. It brings a N=8 SUGRA expert together with Spinfoam experts like Laurent Freidel and John Barrett.

It didn't help to answer about something off topic, did it?

 

[MTd2]

It didn't help to answer about something off topic, did it?

It did :) . But I think you answer is on topic if posted on this thread, so... I will repost my post #128 there :)

 

[marcus]

With any luck, sometime soon you can read this paper on the arXiv:

Aristide Baratin and Laurent Freidel
Hidden quantum gravity in 4d Feynman diagrams: emergence of spin foams

The idea is that any ordinary quantum field theory in 4d Minkowski spacetime can be reformulated as a spin foam model. This spin foam model is thus a candidate for the G -> 0 limit of any spin foam model of quantum gravity and matter!

In other words, we now have a precise target to shoot at. We don't know a spin foam model that gives gravity in 4 dimensions, but now we know one that gives the G -> 0 limit of gravity: i.e., ordinary quantum field theory. So, we should make up a spin foam model that reduces to Baratin and Freidel's when G -> 0.

The fascinating thing I noticed is that their spin foam model seems to be based on the Poincare 2-group. I invented this 2-group in my http://www.arxiv.org/abs/hep-th/0206130" . The physical meaning of their spin foam model was unclear, and some details were not worked out, but it was very tantalizing. What did it mean?

I now conjecture - and so do Baratin and Freidel - that when everything is properly worked out, Crane and Sheppeard's spin foam model is the same as Baratin and Freidel's. So, it gives ordinary particle physics in Minkowski spacetime, at least after matter is included (which Baratin and Freidel explain how to do).

If this is true, one can't help but dream...

... that deforming the Poincare 2-group into some sort of "quantum 2-group" could give a more interesting spin foam model: ideally, something that describes 4d quantum gravity coupled to matter! This more interesting spin foam model should reduce to Baratin and Freidel's in the limit G -> 0.

Of course this dream sounds "too good to be true", but there are some hints that it might work, to be found in http://arxiv.org/abs/hep-th/0501191" . In particular, they describe gravity in way (equation 26) which reduces to BF theory as G -> 0.

Optimistic hopes in quantum gravity are usually dashed, but stay tuned.

This is the initial post of this thread. What it talks about is stuff that didn't really take shape until just now. Baez has posted a 2-groups paper he did with Laurent Freidel, Aristide Baratin, and Derek Wise.

http://math.ucr.edu/home/baez/2rep.pdf
Infinite-Dimensional Representations of 2-Groups
John Baez, Aristide Baratin, Laurent Freidel, Derek K. Wise

AFAIK this paper was posted 23 December 2008. But it is somehow a continuation of what Baez said at the beginning of this thread, in June 2006

 

[MTd2]

Marcus, in that article, a applications in spim foams are proposed as possible aplications:

"We conclude with some possible avenues for future investigation. First, it will be interesting to study examples of the general theory described here. Representations of the Poincare 2-group have already been studied by Crane and Sheppeard [14], in view of obtaining a 4-dimensional state sum model with possible relations to quantum gravity. Representations of the Euclidean 2-group (with G = SO(4) acting on H = R4 in the usual way) are somewhat more tractable. Copying the ideas of Crane and Sheppeard, this 2-group gives a state sum model [7, 8] with interesting relations to the more familiar Ooguri model."

[7] A. Baratin and L. Freidel, Hidden quantum gravity in 4d Feynman diagrams, Class. Quant. Grav. 24 (2007), 2027-2060. Also available as http://arXiv.org/pdf/hep-th/0611042.

[8] A. Baratin and L. Freidel, State-sum dynamics of at space, in preparation.

[14] L. Crane and M. D. Sheppeard, 2-Categorical Poincare representations and state sum applications,
available as arXiv:math/0306440.

There are other references that shows that shows this approach will soon find concreteapplication.