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

[john baez]

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.

 

[f-h]

Finally, you've been teasing us with remarks about this work for some time now ;)

 

[francesca]

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

We are waiting for...
meanwhile I call back the former paper:

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 Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare 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."

 

[marcus]

some of us should probably learn a bit about the Poincaré 2-group
does anyone have one or more of Baez TWF to recommend?
there ought to be one that is ideal to get started understanding about 2-groups and the Poinc in particular

meanwhile I will expand out the references JB gave us just now

http://www.arxiv.org/abs/math.QA/0306440
2-categorical Poincare Representations and State Sum Applications
L. Crane, M.D. Sheppeard
16 pages, 1 figure

"This is intended as a self-contained introduction to the representation theory developed in order to create a Poincare 2-category state sum model for Quantum Gravity in 4 dimensions. We review the structure of a new representation 2-category appropriate to Lie 2-group symmetries and discuss its application to the problem of finding a state sum model for Quantum Gravity. There is a remarkable richness in its details, reflecting some desirable characteristics of physical 4-dimensionality. We begin with a review of the method of orbits in Geometric Quantization, as an aid to the intuition that the geometric picture unfolded here may be seen as a categorification of this process."

http://www.arxiv.org/abs/hep-th/0206130
Higher Yang-Mills Theory
John C. Baez
20 pages

"Electromagnetism can be generalized to Yang-Mills theory by replacing the group U(1)$ by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form electromagnetism to a kind of 'higher-dimensional Yang-Mills theory'. It turns out that to do this, one should replace the Lie group by a 'Lie 2-group', which is a category C where the set of objects and the set of morphisms are Lie groups, and the source, target, identity and composition maps are homomorphisms. We show that this is the same as a 'Lie crossed module': a pair of Lie groups G,H with a homomorphism t: H -> G and an action of G on H satisfying two compatibility conditions. Following Breen and Messing's ideas on the geometry of nonabelian gerbes, one can define 'principal 2-bundles' for any Lie 2-group C and do gauge theory in this new context. Here we only consider trivial 2-bundles, where a connection consists of a Lie(G)-valued 1-form together with an Lie(H)-valued 2-form, and its curvature consists of a Lie(G)-valued 2-form together with a Lie(H)-valued 3-form. We generalize the Yang-Mills action for this sort of connection, and use this to derive "higher Yang-Mills equations". Finally, we show that in certain cases these equations admit self-dual solutions in five dimensions."

 

[Kea]

Hi John and PF friends

Well, I've been dying to know what you guys are up to over there! Unfortunately, I was just in hospital for a week (but don't worry, I'm quite OK), and now my family are forcing me to relax in Oz and my computer time is a little limited.

I don't think it's overly optimistic to hint at great hopes at this point...but then I always err on the side of enthusiasm. We should also be seeing some papers by Laurent/Artem soon, no?

 

[marcus]

... I was just in hospital for a week (but don't worry, I'm quite OK), and now my family are forcing me to relax in Oz and my computer time is ...

sorry to hear that! hope you are all better soon!
it is nice your family cares but a nice wireless laptop with full-color flat screen is better than flowers and much more relaxing. don't the dears understand that plenty of time on the web is essential for convalescence?

 

[Kea]

...but a nice wireless laptop with full-color flat screen is better than flowers...

Yes, Marcus! And guess what? They are going to get me one! A laptop, I mean. I feel so spoilt. I can't wait.

 

[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 a good clue, amigos, I hope we can pay some attention and catch the drift of it.

I believe there is a connection with a friend of a friend R. Brown who wrote the basics about "crossed modules" (correct me if I am wrong, that covers what I'm about to say also).

It appears that Baez invented LIE 2-groups and therefore probably invented LIE crossed modules. and so he was the first to talk about the POINCARÉ 2-group. But the general theory of (non-Lie) 2-groups and crossed modules had already been developed by others----and he cites a 1976 R.Brown paper on some of that.

the hints I am getting are very simple
A. I might need to understand Lie 2-groups (or at least one special case the Poincaré case) to understand 4D gravity
and from there to move on to the QUANTUM DEFORMED Poinc 2-group.

B. it would help a lot to simply understand Lie crossed modules because every Lie 2-group is EQUIVALENT to a Lie crossed module-----the easiest way to construct examples is to just consider crossedmodules because they are one-for-one the same as 2-groups

AND CROSS MODULES APPEAR TO BE COMPARATIVELY EASY TO UNDERSTAND!

-------------
if you look at the new stuff we just got from Baez probably the easiest thing to assimilate in the whole batch (if you are like me) is the idea of a Lie Crossed Module. this is refreshing and reassuring. it is not all Krazy Kats and Monky Barz, there is something very natural about the idea.

even philistines like me can love crossed modules.

then maybe that can be my kitchen stool or stepladder to understand the Poincaré 2-group.

and for some people it may be more intuitive for them to jump right to the 2-group, skipping this first step

so I will try to write a post here about crossed modules, just paraphrasing. (help, anybody who wants)

 

[marcus]

first a bit about 2groups to motivate
everybody can picture a Lie group like the group of all rotations of a sphere of some dimension or some other group of symmetries or motions

so picture a nice Lie group as say a nice naked classic torso of some Roman emperor whose name has been forgotten----and now imagine it clothed in CHAIN MAIL. every point has a little ring sewn on it.

now the RING is a wellknown group and you can label it in lots of different ways by complex numbers and have it spin itself around by multiplying. You can have MILLIONS OF DIFFERENT group morphism mappings between these rings that map-wrap ring A around ring B some number of times. So amongst this swarm of little ring groups there is a huge buzzing hive of algebraic possibilities, algebraically morphing one ring to another.

So now we have a GROUP OF GROUPS! the original staid old torso, the roman emperor or maybe it was just a famous general, WAS A GROUP and now this group is made up of a huge buzzing swarm of little groups.

and these are all Lie----which is to say the smooth-type group that you can take derivatives in and find tangents to and stuff.

well that is probably enough for one post

 

[marcus]

Now I just went ahead and DID what he suggested and replaced every occurence of word "set" by "Lie group"
===MODIFIED quote Baez===
The concept of ‘Lie 2-group’ is a kind of blend of the concepts of Lie group and category. A small category C has a Lie group C₀ of objects, a Lie group C₁ of morphisms, functions s, t: C₁-> C₀ assigning to each morphism f: x -> y its source x and target y, a function i: C₀ -> C₁ assigning to each object its identity morphism, and finally, a function

$$\circ: C_1 \times_{C_0} C_1 \rightarrow C_1$$

describing composition of morphisms, where

$$C_1 \times_{C_0} C_1 = \{(f, g) \in C_1 \times C_1 : t(f) = s(g)\}$$ is the Lie group of composable pairs of morphisms. If we now take the word ‘function’ and replace it by ‘homomorphism’, we get the definition of a Lie 2-group.

Here and in all that follows, we require that homomorphisms between Lie groups be smooth:
=====endquote====

Now we get the official definition , still on page 8
===quote Baez===

Definition 1. A Lie 2-group is a category C where the set C₀ of objects and the set C₁ of morphisms are Lie groups, the functions s, t: C₁ -> C₀ , i: C₀ -> C₁ are homomorphisms,

$$C_1 \times_{C_0} C_1$$ is a Lie subgroup of $$C_1 \times C_1$$ ,

and $$\circ: C_1 \times_{C_0} C_1 \rightarrow C_1$$
is a homomorphism. The fact that composition is a homomorphism implies the exchange law

$$(f_1 \circ f'_1)(f_2 \circ f'_ 2) = (f_1f_2) \circ (f'_1f'_2)$$
=========endquote======
the wonder is that this actually say fairly natural, you might say obvious or trivial, stuff. After a while it seems like the obvious things to require. something's right. just takes a while to get used to

 

[marcus]

heh heh that image of a classical torso in chain mail is almost certainly wrong, but i won't erase it for now at least. the main thing is we got up to page 9 in the Baez hep-th/0206130
so now we are looking at LIE CROSSED MODULES which I suspect are a good thing. and we have the motivation for them that they correspond one-to-one with Lie 2-groups.

my wife and I are listening to the Mozart Cminor which she thinks she would like to learn to sing part of with a local community chorus, and its late, so I won't try to do more with this tonight. more tomorrow

 

[Chronos]

Dr. Baez: I think Louis Crane has some very interesting ideas. His contribution, and implied endorsement, has me all ears. I hope you don't mind if I mention [perhaps you did and I overlooked it] this paper of yours:

From Loop Groups to 2-Groups
http://www.arxiv.org/abs/math.QA/0504123

 

[marcus]

Beautiful morning----sometimes Berkeley has the feel of being in the tropics, like on a Caribbean island. Looks like its going to be a hot day.
Last night I ACCIDENTALLY ERASED a post between what is #9 and #10 which is OK. Let's make a fresh start.

And eventually Hurkyl or selfAdjoint might take over, so this might just be an interlude or sideshow.

the main thing. the definition of a Lie 2-group is amazingly almost ridiculously NATURAL. the tamest possible defintion of a CATEGORY is the "small" version where the classes of objects and morphisms are sets---this is just how you or I would picture a category so it is a formality really. And what Baez says to do is TAKE THE DEF OF CAT AND SIMPLY REPLACE THE WORD SET BY THE WORD LIE GROUP! nothing easier and also also replace the idea of a simple function between sets by the identity-and-multiplication-preserving homomorphism between groups---homz is just the natchrul mapz between groupz.

hey people, we know that both Lie groups and categories are hugely historically important, they are like TECTONIC in physics and math. so people are clowns if they do not put the ideas together, which is what Baez is doing here and he calls the result a Lie 2-group.

===MODIFIED quote Baez===
The concept of ‘Lie 2-group’ is a kind of blend of the concepts of Lie group and category. A small category C has a set C₀ of objects, a set C₁ of morphisms, functions s, t: C₁-> C₀ assigning to each morphism f: x -> y its source x and target y, a function i: C₀ -> C₁ assigning to each object its identity morphism, and finally, a function

$$\circ: C_1 \times_{C_0} C_1 \rightarrow C_1$$

describing composition of morphisms, where

$$C_1 \times_{C_0} C_1 = \{(f, g) \in C_1 \times C_1 : t(f) = s(g)\}$$ is the set of composable pairs of morphisms.If we now take the words 'set' and ‘function’ and replace them by 'Lie group' and ‘homomorphism’, we get the definition of a Lie 2-group.
Here and in all that follows, we require that homomorphisms between Lie groups be smooth:
=====endquote====

now I will do the suggested copy and paste to replace the words in the above quote

===MODIFIED quote Baez===
The concept of ‘Lie 2-group’ is a kind of blend of the concepts of Lie group and category. A Lie 2-group C has a Lie group C₀ of objects, a Lie group C₁ of morphisms, homomorphisms s, t: C₁-> C₀ assigning to each morphism f: x -> y its source x and target y, a homomorphism i: C₀ -> C₁ assigning to each object its identity morphism, and finally, a homomorphism

$$\circ: C_1 \times_{C_0} C_1 \rightarrow C_1$$

describing composition of morphisms, where

$$C_1 \times_{C_0} C_1 = \{(f, g) \in C_1 \times C_1 : t(f) = s(g)\}$$ is the Lie group of composable pairs of morphisms.

If we now take the word ‘function’ and replace it by ‘homomorphism’, [WHICH I NOW JUST DID BY CUT AND PASTE] we get the definition of a Lie 2-group.

Here and in all that follows, we require that homomorphisms between Lie groups be smooth:
=====endquote====

ALL THAT STUFF WITH the maps s,t, and i is just formalities that you have to have in a category. If you think about it categories is just a setup where you have maps between objects and so for every map you need something that tells you the SOURCE and the TARGET of that map, where it leaves from and where it goes to. and the axiom of category says that every object has a special distinguished map from itself to itself which is the IDENTITY map, so you need something that associates to each object its identity "morphism" (thats the word for map). So all this stuff he's telling us is just trivial natural STRUCTURE that you have to have. Oh and there is the composition of morphisms business where you go by one map and then proceed on your way with another, which means the target of the first must be the same as the source of the next, simply so they connect.

And he says let's take this obvious structure and make it all LIE smooth.
Lie groups are just groups with some built-in smoothness. So this is the incredibly obvious merging of two tectonic things, where only the most obvious structure is being required to be smooth.

Now we get the official definition , still on page 8
===quote Baez===

Definition 1. A Lie 2-group is a category C where the set C₀ of objects and the set C₁ of morphisms are Lie groups, the functions s, t: C₁ -> C₀ , i: C₀ -> C₁ are homomorphisms,

$$C_1 \times_{C_0} C_1$$ is a Lie subgroup of $$C_1 \times C_1$$ ,

and $$\circ: C_1 \times_{C_0} C_1 \rightarrow C_1$$
is a homomorphism. The fact that composition is a homomorphism implies the exchange law

$$(f_1 \circ f'_1)(f_2 \circ f'_ 2) = (f_1f_2) \circ (f'_1f'_2)$$
=========endquote======

s and t go from the morphisms to the objects, because for each morphism they tell you its source and its target object.

and i goes from the objects to the morphisms, because for each object it picks out for you the morphism which is the identity morphism on that object.

the composition axiom with the little circle simply says that composition of morphisms is defined where you expect it to be, namely where the target destination of the first is the same as the source point of departure of the second. (the next train leaves from the station where you just arrived) (in Chicago you sometimes have to take a taxi because they have two trainstations and stupidly enough your outbound train might leave from a different station from where you are so watch out and be careful if you are ever in Chicago)

Finally there is the EXCHANGE LAW which says that Lie group multiplication can serve as a stand-in for composition of mappings (do one and then the other, connecting flights)
This is the beautiful thing here (there had to be at least one beautiful thing). It is that BAEZ IS MAKING THE MORPHISMS OF THIS CATEGORY INTO A LIE GROUP and we know that morphisms COMPOSE---i.e. you can go by first one and then connect and go by the second one----and composition is a lot like MULTIPLICATION---in fact the root idea of multiplying the way that scaling operations in the real world compose, you can scale something up by 2 and then scale it up by 3 and that has the effect of scaling it up by 6.
So if the experiment is going to work, group multiplication will have to switch-hit with composition, they have to be alter egos of each other.

note that somebody else invented the idea of a 2-group----which would have had this "exchange law" requirement----and what Baez is now doing is making it SMOOTH. (and also another important thing he is doing is making it INTERESTING by connecting it to the study of spacetime, which otherwise it would be merely pure mathematics) creativity comes in different shapes and sizes. this whole thing has me bugeyed with excitement.

 

[john baez]

A beautiful day for Lie 2-groups

Beautiful morning----sometimes Berkeley has the feel of being in the tropics, like on a Caribbean island. Looks like its going to be a hot day.

It's hot up here in Waterloo, Canada!

the main thing. the definition of a Lie 2-group is amazingly almost ridiculously NATURAL. the tamest possible defintion of a CATEGORY is the "small" version where the classes of objects and morphisms are sets---this is just how you or I would picture a category so it is a formality really. And what Baez says to do is TAKE THE DEF OF CAT AND SIMPLY REPLACE THE WORD SET BY THE WORD LIE GROUP!

Yes, that's all! And you're even using enough capital letters to convince me that the flash of realization is hitting you just like it hit me: This stuff is simple and sweet! Why in the world hasn't everyone already studied the heck out of it??

But, just for fun, let me try to say what you said another way.

I assume that with Kea around, everyone here knows what a category is.

It's a gadget with "objects":

x

and "morphisms" between objects:

x --f--> y

You can compose a morphism from x to y with one from y to z:

x ---f--> y --g--> z

and get one from y to z.

Composition is associative, and every object has an identity morphism.

That's it!

Now, what's a 2-group? It's the same sort of thing, but now the
objects form a group, so you can multiply them: if you have x and
x', you can multiply them and get

xx'

and also the morphisms form a group, so you can multiply them!
If you have

x --f--> y

and

x' --f'--> y'

you can multiply them and get

xx' --ff'--> yy'

There's just one more thing: composing arrows gets along with
multiplying arrows. In other words

xx --ff'--> yy' --gg'--> zz'

is the same as what you get by multiplying

x --f--> y --g--> z

and

x' --f'--> y' --g'--> z'.

That's it! I haven't left anything out.

And now, if our group of objects and our group of morphisms are
Lie groups, and all our operations are smooth, we say we have a Lie 2-group.

In the Poincare 2-group, the group of objects is the group of Lorentz
transformations, and the group of morphisms is the Poincare group.

That doesn't completely describe the Poincare 2-group. You need to
know some other stuff, like:

If you have a morphism in here, which object does it start at, and
which object does it end at?

There's only one sensible answer to this question, I think, so I'll leave it as a puzzle.

You also need to decide how to compose morphisms. I'll leave that as a (harder) puzzle.

 

[selfAdjoint]

If you have a morphism in here, which object does it start at, and
which object does it end at?

There's only one sensible answer to this question, I think, so I'll leave it as a puzzle

I have been trying to work out this puzzle, guided by two principles:
1) The answer to this kind of thing is always a "DUH!". You are creating a building block for future construction and because of the second principle below, any extra complication you introduce now will come back and bite you. So the answer will seem astoundingly trivial when it is shown.

2)Everything that is not forbidden is compulsary.


BWT, Marcus, do you have a source for this slogan? My own sources are T.H. White, who use it as the motto of the ant hill in The Sword in the Stone. and L. Sprague de Camp, who used the German translation, with clauses reversed ("Alles was nicht Pflicht ist, ist verboten") as the motto of the corporate state in his sf novella The Stolen Dormouse. Both of these uses are from about 1939. Did White and de Camp, on opposite sides of the Atlantic, have a common source?

So the objects are elements of the Lorentz group; special orthogonal transformation on spacetime, i.e. rotations and Lorentz boosts. Let me call them twists. And the morphisms are elements of the Poincare Group, which is the Lorentz group plus the translations, so a typical morphism is a twist times a shift. So how would a typical (twist X shift). say (ts), act of a typical "object twist x? Why by mapping x into t! Duh! This, however leaves the shifts out in the cold. Some of the elements of the Poincare group are pure shifts, and they should be morphisms too. I think they map everything into the identity element. That would work with a product morphism like (ts), you could define the rule as IM(t X s) (x) = IM(t)(x) X IM(s)(x)) = t X e = t in the L group

How'm I doing so far?

 

[john baez]

I have been trying to work out this puzzle, guided by two principles:

1) The answer to this kind of thing is always a "DUH!". You are creating a building block for future construction and because of the second principle below, any extra complication you introduce now will come back and bite you. So the answer will seem astoundingly trivial when it is shown.

Right. You understand the way these things work... but you made a little slip, which prevented you from guessing the truly simple solution:

So the objects are elements of the Lorentz group; special orthogonal transformation on spacetime, i.e. rotations and Lorentz boosts. Let me call them twists. And the morphisms are elements of the Poincare Group, which is the Lorentz group plus the translations, so a typical morphism is a twist times a shift. So how would a typical (twist X shift). say (ts), act of a typical "object" twist x? Why by mapping x into t! Duh!

No, it's even simpler: the group of all morphisms is the Poincare group, so a typical element is of the form ts, and this element is a morphism from t to itself.

I think your mistake was treating a morphism as a function which sends any object x into some other object. Actually, in a category, any morphism

x --f--> y

has a single object x as its source, and a single object y as its target.

A morphism ts in the Poincare 2-group consists of a twist t and a shift s. Its source is a twist - which you have to guess somehow - built from t and s. Its target is another twist built from t and s. The simplest answer is to use t for both source and target!

Simple enough?

 

[selfAdjoint]

I am a little confused about the categorization of a group. The category is supposed to have one object, denoted *, and the group elements are the morphisms. But take the simple group of two elements, e and a, with the multiplication table ee = aa = e, ae = ea = a. I understand how you get an identity morphism; e: * -> * by e(*) = *. But how do you define the non-identity elemental morphism a:* -> * ?

 

[f-h]

e(*) = *

I think this is wrong or at least highly misleading. Remember that a morphism is defined by itself and not by the way it maps objects to objects. So you get a(*) = * as well (where I take that string of symbols to read that * is the target of the morphism a which has * as it's source), yet they are distinct, and their composition (one is inclined to write a(e (*)), thinking of morphisms as functions, which they are not) is given by the multiplication table.

Was that the question?

 

[john baez]

I am a little confused about the categorization of a group.

I'm not sure what you mean by "categorization" here. I like to talk about a process called http://front.math.ucdavis.edu/math.QA/9802029" , where we take ordinary math based on sets and replace the sets by categories. If we categorify the concept of "group" we get the concept of 2-group, which I defined a while back on this thread.

On the other hand, a group is already a kind of category! A group can be regarded as a category with one object, with all its morphisms being invertible.

I haven't been using that perspective in talking about the Poincare 2-group in this thread, since I wanted to keep everything as familiar and unthreatening as possible.

But, you seem to be invoking it here:

The category is supposed to have one object, denoted *, and the group elements are the morphisms. But take the simple group of two elements, e and a, with the multiplication table ee = aa = e, ae = ea = a. I understand how you get an identity morphism; e: * -> * by e(*) = *.

You're again making that mistake of treating a morphism as a "function" from objects to objects. It's not, so the equation e(*) = * makes no sense.

A morphism is just an abstract thing f with some object x called its source and some object y as its target. To say this quickly, we write f: x -> y. That's all! It's simpler than you're imagining.

But how do you define the non-identity elemental morphism a:* -> * ?

There's nothing to "define": it's not as if you're defining a function of some sort. A morphism is not a function from objects to objects.

So, to think of this group as a category, you just say there's one object * and two morphisms,

e: * -> *

and

a: * -> *.

You have to say how to compose the morphisms a and e in various ways, but that's easy - you just use the group multiplication table.

By the way, if I wanted to categorify this fact:


A group is a category with one object and with all morphisms invertible.

I would say this:


A 2-group is a 2-category with one object and with all morphisms and 2-morphisms invertible.

This is indeed true, and it's http://front.math.ucdavis.edu/math.QA/0307200" . But since far fewer people are comfy with 2-categories than with categories, in this thread I gave a definition of "2-group" that only mentions categories, not 2-categories - just as one can define "group" while only mentioning sets, not categories!

So, I'm not completely sure why you brought up the fancier approach... but it's all good stuff.

 

[selfAdjoint]

Thank you, this is clear to me now.

 

[marcus]

It's hot up here in Waterloo, Canada!
...

In the Poincare 2-group, the group of objects is the group of Lorentz
transformations, and the group of morphisms is the Poincare group.

That doesn't completely describe the Poincare 2-group. You need to
know some other stuff, like:

If you have a morphism in here, which object does it start at, and
which object does it end at?

There's only one sensible answer to this question, I think, so I'll leave it as a puzzle.

You also need to decide how to compose morphisms. I'll leave that as a (harder) puzzle.

About the puzzle---I think selfAdjoint has already figured it out but I did not follow all the discussion, so I will take a guess.

Someone, I think it was JB, suggested using letters T and S for "twist" and "shift", where T is an element of the Lorentz group and S is thought of as a translation (we are building the Poincaré group)
[SNIP]

[EDIT] GROAN. I just looked at the "Higher Yang Mills" paper that I lost a couple of days ago, while cleaning up the living room. the Poincaré 2-group is explained as EXAMPLE 9. So the puzzle was already answered.
[/EDIT]

...I talked to Kea and she suggested one by Girelli and Pfeiffer, which I never got around to looking at it. Maybe I will now.

http://arxiv.org/abs/hep-th/0309173
Higher gauge theory -- differential versus integral formulation
Florian Girelli, Hendryk Pfeiffer
26 pages
DAMTP-2003-86
J.Math.Phys. 45 (2004) 3949-3971
"The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of non-Abelian groups and generalizes the formulation of gauge theory in terms of parallel transports..."

I hope Kea is better now and was wishing she would suddenly materialize amidst this thread.

Well, looking at Girelli/Pfeiffer, I see right away references
[15] J. C. Baez: Higher Yang–Mills theory (2002). Preprint hep-th/0206130.
[19] J. C. Baez and A. Crans: Higher dimensional algebra VI: Lie 2-algebras (2003). Preprint math.QA/0307263.

My posts are just not helpful in this thread at this point. I shall delete the next one.

 

[selfAdjoint]

This is an excerpt from TWF week 86:

You may or may not remember, but in "week80" I explained the idea of a
"2-category" pretty precisely. This is a gadget with a bunch of
objects, a bunch of morphisms going from one object to another, and a
bunch of 2-morphisms going from one morphism to another. We write i
f: x -> y to denote a morphism f from the object x to the object y, and we
write F: f => g to denote a 2-morphism F from the morphism f to the
morphism g.

So a 2-category has the usual morphisms, which just amount to ordered pairs of objects (I am clear on that now), plus it has 2-morphisms which go from one morphism to another, i.e. ordered pairs of elements of Hom(obj). And in a 2-group, all these morphisms and 2-morphisms are invertible.

I was the one who introduced twist and shift; a twist is an element of the Lorentz group, a special orthogonal transformation on spacetime, i.e. a boost or a rotation, and a shift is a translation. The general element of the P-group can be represented as a product ts of a twist and a shift.

 

[marcus]

thanks!

do you have some intuition to share about the "crossed module" way of writing a twogroup?

I think I found a couple of typos on page 10 of "Higher Y-M" that confused me earlier
at the bottom of the page, in example 6, shouldn't rho be alpha
and shouldn't t:G -> H really be t -> G?

with those changes, then example 9 (which says "as in example 6" at one point) becomes easier to grasp
(I think, I hope)

 

[john baez]

whoops!

do you have some intuition to share about the "crossed module" way of writing a twogroup?

Yes. I know you weren't asking me... but the crossed module way of thinking about a 2-group is:

1) a very convenient way to "slice and dice" the concept of 2-group into pieces that can be managed without knowledge of category theory,

2) potentially misleading because it chops a concept with its own inherent integrity (a category in the world of groups!) into a seemingly arbitrary list of data (a group G, a group H, an action of G on H, and a homomorphism from H to G, satisfying two axioms),

3) very handy for higher gauge theory, because a "2-connection" consists of a 1-form A valued in the Lie algebra of G, which describes parallel transport along paths, together with a 2-form B valued in the Lie algebra of H, which describes parallel transport along surfaces,

4) easiest to understand with lots of pictures, some of which I drew at the beginning of http://arxiv.org/abs/hep-th/0206130" .

5) impossible to really understand without also understanding the more categorical approach to 2-groups, especially since this motivates the pictures.

An element g of G describes a process, or transformation, or symmetry. An element of H describes a process-between-processes, or transformation-of-transformations, or symmetry-between-symmetries, which starts at the identity:

1 --h-> g

It just so happens that starting from these one can build up the general processes-between-processes, which start at any element of G and end at any other element of G:

g --f--> g'

In my quick and easy definition of 2-groups earlier in this thread, I spoke about the general processes-between-processes, and called them "morphisms" - that's what they are, if we think of a 2-group as a category in the world of groups.

I think I found a couple of typos on page 10 of "Higher Y-M" that confused me earlier
at the bottom of the page, in example 6, shouldn't rho be alpha

Yup, if I was using alpha as the name for how G acts on H.

and shouldn't t:G -> H really be t -> G?

Yup.

with those changes, then example 9 (which says "as in example 6" at one point) becomes easier to grasp (I think, I hope).

I imagine so! Maybe I'll fix this someday... I decided to never publish "Higher Yang-Mills Theory", because there are certain things I don't like about it; all the information here will find its way into http://www.arxiv.org/abs/math.QA/0307200" . But, it has the advantage of being short and sweet.

 

[john baez]

So a 2-category has the usual morphisms, which just amount to ordered pairs of objects (I am clear on that now),

A morphism is not just an ordered pair of objects: a morphism determines an ordered pair of objects, called its source and target, but it has more information. If a morphism f has source x and target y, we write

f: x -> y

for short, but do not think that the morphism f simply is the ordered pair (x,y).

Consider the category Set. Here objects are sets and morphisms are functions. A function f determines two sets x and y called its source and target (or "domain" and "codomain"), and then we write f: x -> y. But, lots of different functions have the same source and target, so a function is not just an ordered pair of sets.

plus it has 2-morphisms which go from one morphism to another, i.e. ordered pairs of elements of Hom(obj).

Same problem: a 2-morphism determines an ordered pair of morphisms, called its source and target, but one should not say it is this ordered pair.

And in a 2-group, all these morphisms and 2-morphisms are invertible.

Yup. Here's how I said it, a while back:

It's a gadget with "objects":

x

and "morphisms" between objects:

x --f--> y

You can compose a morphism from x to y with one from y to z:

x ---f--> y --g--> z

and get one from y to z.

Composition is associative, and every object has an identity morphism.

That's it!

Now, what's a 2-group? It's the same sort of thing, but now the
objects form a group, so you can multiply them: if you have x and
x', you can multiply them and get

xx'

and also the morphisms form a group, so you can multiply them!
If you have

x --f--> y

and

x' --f'--> y'

you can multiply them and get

xx' --ff'--> yy'

There's just one more thing: composing arrows gets along with
multiplying arrows. In other words

xx --ff'--> yy' --gg'--> zz'

is the same as what you get by multiplying

x --f--> y --g--> z

and

x' --f'--> y' --g'--> z'.

 

[selfAdjoint]

Yes, after I wrote "ordered pair" I realized some of what you posted here and thought I should go back and edit it to "labelled ordered pair" at least (I think that would meet you objections?) And you'll see on the other thread Marcus started that I did that. (Added in edit: I see this morning that that post didn't make it through. My home computer has been having problems. The post responded to Mike2's #4 on that thread and concerned your TWF 86, petal diagrams , and the definition of 2-group). I have a visual imagination and I liked your "daisy" image for the 1-group and 2-group.

What I need to get my head around is what 2-morphisms do, besides their definition. I have an intuition that they make commutative diagrams simple. The paper says that the way the Poincare group splits into the Lorentz transformations and translations can't be expressed naturally in 1-categories but can in terms of 2-groups.

 

[Dcase]

Nucleic acids as transformation - morphisms?

This speculation may or may not be related to this thread - hopefully there is some insight with using helices and loops [zero helical angle] as a category:
Eventually all gauges from QM to GR will need to be linked or unified through either loops or strings or both if not some other method .
Helical strings offer an advantage since EM in phasors, particles in Schroedinger waves and planetary mechanics all have trajectories in common with rifled gun ballistics.
The helix may be the geodesic of space-time.
[I could easily be over interpreting David Hestenes [physics emeritus, ASU] ‘The Kinematic Origin of Complex Wave Functions’ that appears to apply to across all gauges of QM and GR physics and perhaps to macromolecular gauges.]
The coiled helix may even suffice for the concepts of large and small unseen curled-up dimensions.
A visual aid may be easily had by substituting gauge and period appropriate springs or slinky-toys for the loops in the Rovelli image.
http://www.cpt.univ-mrs.fr/~rovelli/loop_quantum_gravity.jpg

There exists within biophysics and biochemistry a Modulus-2 system that has [orthogonal?] transformations which are:
1 - “a boost or a rotation” that is known as transcription
2 - “and a shift is a translation” that is known as translation
in the nucleic acid transformation of DNA through mRNA into amino acid proteins.

Modulus-2 is a three-number [0,1,2] system in helical mode, but either of two, two-number systems in loop [circle or ellipse] mode, with either the leading or trailing digit omitted.

Loop Modulus-2:
1 - The [0, 1] or [off, on] binary is used to interface human with computer language.
This may be the binary used in E = mc^2,
with 0 representing no mass or all energy or an annihilation transformation
and 1 representing all mass or no energy or a creation transformation.

2 - The [1, 2] or [this, that] binary appears to used by nature in a nested manner with nucleic acids.
First level - 1 may represent RNA with 2 as DNA
Second level - 1 may represent pyrimidine with 2 as purine
Third levels - if a pyrimidine then 1 may represent U or T dependent on whether RNA or DNA present with 2 as C
or if a purine then 1 may represent A with 2 as G.

Thus any of 2^2 possible candidates may occupy a position when inserted into the helical string:
1,1 would be pyrimidine U or T dependent upon whether RNA or DNA
1,2 would be pyrimidine C
2,1 would be purine A
2,2 would be purine G.

The transcription transformation would involve complementary symmetry [with errors possible] through a template and mRNA having the relationship [U or T] complement A while C complements G.

Helical Modulus-2
The translation transformation would involve the 2^2^2 possible genetic codes read as a triplet or 3-bit Helical Modulus-2 utilizing 0,1,2 positions which due to helical periodicity can be visualized as 0-Pi, 1-Pi, 2-Pi.
There are actually 64 codes for RNA with U and 64 identical codes for DNA with T.

The genetic code is available in many textbooks or at many websites such as this one from UT-Health Center, Tyler:
http://psyche.uthct.edu/shaun/SBlack/geneticd.html

The primary strand is read from “5' to 3' ("Five prime to three prime").”
The complementary strand from "3' to 5'".

RNA appears to be older than DNA
U appears to be older than T or C.
A appears to be older than G.

The genetic code appears to have evolved in a temperature dependent manner from a couplet to a triplet.

Loops or strings or both or some other method will eventually be a mathematical representations of nucleic acids.
This effort may or may not apply to larger gauges such as GR or smaller gauges such as QM. The physics of a helix should be similar but not identical for different gauges.

Analogy: For the category fusion, similar but not identical processes may include: string joining, nuclear fusion, cell fertilization, black hole mergers.

 

[selfAdjoint]

Well, biological helices are http://scripts.iucr.org/cgi-bin/paper?S0907444905042654"

And quaternions http://www.math.sunysb.edu/~tony/bintet/tetgp.html"

 

[Dcase]

Hi selfAdjoint:

Thanks for the references to
1 - What is the Binary Tetrahedral Group?
2 - Biological Crystallography

Reference #1 links to Quaternionic representation of the Binary Tetrahedral Group
http://www.math.sunysb.edu/~tony/bintet/quat-rep.html

This link has the statement “1 and - 1 complete the list of 24 elements.”

Does this somehow relate to the complex-24D of the Monstrous Moonshine?

Is there an octonion relationship?
http://math.ucr.edu/home/baez/octonions/

‘To Infinity and Beyond’ Eli Maor p35 has the sum
((2^24) * 76 977 927 * (Pi^26)) / (27 !) = 1.00000001
for the series 1/1^k + 1/2^k + 1/3^k + 1/4k + ...

The numbers 24 and 26 appear to have some type of bosonic or other relationship.
Now you show me that there is even a biological relationship!
Is this by accident?

Consider Fe as element #26 [the most common metal in our known universe] used by most fauna as the metal of hemoglobin.
Hemocyanins Cu as element #27 [the second most common metal in the oceans] used by most molluscs and some arthropods.
Chlorophyll Mg as element #12 [the most common metal in the oceans] used by flora.
The set of Eucharya seems to be nearly completely covered by these 3 metals having atomic numbers associated with some of the string theories.

Vafa F-theory uses 12-D.

This may all be happenstance - BUT?

Or it may be like the close but no cigar relationship of
1 / Euler's constant = 1.73245471
with
3^(1/2) = 1.73205081

If loop or string theory could be tested through biology, this might facilitate testing in QM and GR.

 

[Dcase]

Hi selfAdjoint:

The helix [helicoid] seems to be important [an accident?] - in biology, ballistics AC and mechanics.

Fluid mechanics appears to use the helicoid as the primary minimal surface for soap bubbles.
'Surface story: inspired by spiral soap films, mathematicians zero in on a novel, economical, and infinite helix' in Science News 17 Dec 2005 by Ivars Peterson
http://www.sciencenews.org/articles/20051217/bob9.asp

I even speculate about a relationship with the logarithmic spiral.

 

[Kea]

Hi Dcase, Marcus, selfAdjoint, John, Hurkl et al

There's so much happening at once - it's hard to know what to do next. I was sort of hoping that some nice mathematical biologists would take care of Monsters and DNA and all that for us so that we could continue our struggle with simple things like 2-topologies (well, simple for some people, but not for me of course).

I think Hurkl has the right idea! Whitehead was way ahead of his time, but that was a long time ago now. No need to worry too much about crossed modules when everything makes sense in terms of 2-categorical geometry. It's really cool how quickly everybody is figuring this all out. It took me years just to understand a little bit of this stuff.

Anyway, cheers
Kea


P.S. The drugs are doing the trick - all the dreadful bugs that I caught in hospital are going away now, so I should feel 100% again soon.

 

[marcus]

...

P.S. The drugs are doing the trick - all the dreadful bugs that I caught in hospital are going away now, so I should feel 100% again soon.

good! we were probably all a little worried. I'm glad you are mending
and hopefully will soon be back in NZ where Keas belong.

 

[kneemo]

I'm not sure I understand what is going on quite yet, but I'm going to take a crack at an example. I'm going to use the exceptional Jordan algebra $$\mathfrak{h}_3(\mathbb{O})$$ as John has written extensively on its automorphism group $$F_4$$, and is very much an expert on this subject. On the other hand, I think it's a cool structure and it would be great if it could serve as a base for a 2-group. ;)

An element of $$F_4$$ gives an isomorphism from the exceptional Jordan algebra to itself. So I'm looking at the exceptional Jordan algebra as the source and target. In result, $$F_4$$ is my 1-group of morphisms.

Next, I'd like to consider morphisms from $$F_4$$ to $$F_4$$. To be concrete, let's look at isomorphisms from $$F_4$$ to $$F_4$$, which form the group $$Aut(F_4)$$. As $$F_4$$ is a 1-group of morphisms, $$Aut(F_4)$$ appears to be a 2-group of "2-morphisms".

Is my reasoning correct so far?

 

[john baez]

automorphism 2-groups and the M-theory 3-group

I'm not sure I understand what is going on quite yet, but I'm going to take a crack at an example. I'm going to use the exceptional Jordan algebra $$\mathfrak{h}_3(\mathbb{O})$$ as John has written extensively on its automorphism group $$F_4$$, and is very much an expert on this subject. On the other hand, I think it's a cool structure and it would be great if it could serve as a base for a 2-group. ;)

An element of $$F_4$$ gives an isomorphism from the exceptional Jordan algebra to itself. So I'm looking at the exceptional Jordan algebra as the source and target. In result, $$F_4$$ is my 1-group of morphisms.

Next, I'd like to consider morphisms from $$F_4$$ to $$F_4$$. To be concrete, let's look at isomorphisms from $$F_4$$ to $$F_4$$, which form the group $$Aut(F_4)$$. As $$F_4$$ is a 1-group of morphisms, $$Aut(F_4)$$ appears to be a 2-group of "2-morphisms".

What you're calling morphisms and 2-morphisms, I prefer to call objects and morphisms, at least when I'm talking to people who may not know 2-category theory. It doesn't really matter, but I don't want people to become more confused than necessary! So, when I gave my definition of 2-group in this thread, here's what I said:

A category a gadget with "objects":

x

and "morphisms" between objects:

x --f--> y

You can compose a morphism from x to y with one from y to z:

x ---f--> y --g--> z

and get one from y to z.

Composition is associative, and every object has an identity morphism.

A 2-group is the same sort of thing, but now the
objects form a group, so you can multiply them: if you have x and
x', you can multiply them and get

xx'

The morphisms also form a group, so you can multiply them!
If you have

x --f--> y

and

x' --f'--> y'

you can multiply them and get

xx' --ff'--> yy'

There's just one more thing: composing arrows gets along with
multiplying arrows. In other words

xx --ff'--> yy' --gg'--> zz'

is the same as what you get by multiplying

x --f--> y --g--> z

and

x' --f'--> y' --g'--> z'.

So, translating into this notation, you seem to be looking for a 2-group with F4 as objects and Aut(F4) as morphisms... or something like that. But you aren't using anything special about the group F4 yet, so we might as well keep things simple and use a general group G. Let's do that.

Is my reasoning correct so far?

Well, you haven't gotten far enough for me to say for sure!

We can try to build a 2-group with G as the group of objects and Aut(G) (or something like that) as the group of morphisms. But there is more left to do:

1) given a morphism f, we need to say what it's source and target are, so we can write it as

x --f--> y

for objects x and y, called its source and target.

2) given morphisms

x --f--> y

and

y --g--> z

we need to say how to compose them and get a morphism

x --f--> y --g--> z

which we could call f o g for short.

3) We need to check that this stuff gets along with multiplication:
multiplying the morphisms

x --f--> y

and

x' --f'--> y'

we need to get a morphism

xx' --ff'--> yy'

and we need to check that the composite of a product of morphisms
is the product of their composites:

xx --ff'--> yy' --gg'--> zz'

is the same as what you get by multiplying

x --f--> y --g--> z

and

x' --f'--> y' --g'--> z'.

So, you should try to do this stuff.

The closest thing I know to what you're describing is the automorphism 2-group AUT(G) of a group G. This is a very natural gizmo - you can read about it in Example 10 http://fr.arxiv.org/abs/hep-th/0206130".

But, AUT(G) has the group Aut(G) as its objects, not as its morphisms!

And, given objects x and y, a morphism

x --g--> y

is an element of g that conjugates x to give y: in other words,

g x(h) g^{-1} = y(h)

for all h in G.

It's a fun idea to see if you can cook up a 2-group with G as objects and Aut(G) (or something like it) as morphisms. I don't know if you can. I know a lot about automorphism 2-groups, though.

If you like the idea of blending exceptional algebraic structures and higher group theory, you may like http://golem.ph.utexas.edu/string/archives/000840.html" . I want to think about this more!

 

[selfAdjoint]

So (once more into the breach...) in the Lorentz/Poincare example, the objects that you can multiply are the rotations and boosts ("twists") for the Lorentz group, and the morphisms you can multiply come from the elements of the Poincare group ("twist-shifts"), and you said that a twist-shift ts defines a morphism from object t to object t.

So now my question is, why do we say the morphisms are described by the Poincare group; why not just the Lorentz group?