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

[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?

 

[john baez]

Do the twist! Do the shift! Do the Poincare 2-group shuffle!

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?

I understand your puzzlement. But maybe you should have said "why not just the translation group"? The objects are Lorentz transformations; to specify a morphism we also need a translation.

The point is this: the set of morphisms from any object to itself is just the translation group, but the set of all morphisms is the Poincare group!

Consider the set of all morphisms. Any morphism looks like
this:

t --ts--> t

To specify it, we need to give the pair (t,s), which is an element of the Poincare group.

Next fix an object t, and consider the set of morphisms from t to itself. Any such morphism looks like this:

t --ts--> t

Just like before! But now we've fixed t ahead of time, so the only thing we get to choose is s, which is an element of the translation group.

In the crossed module approach to 2-groups, we get the Poincare 2-group by taking G = Lorentz group and H = translation group, with the obvious action of G on H. Then the group of all morphisms is the semidirect product of G on H, which is the Poincare group.

In other words, as someone said a while back, we're thinking about the Poincare group as built out of Lorentz transformations (twists) and translations (shifts) in the usual way. But now, we're thinking of the twists as objects and the shifts as automorphisms (morphisms from an object to itself).

 

[john baez]

a cool 2-group

Here's a simple example of a 2-group taken from my discussions with http://www.dcorfield.pwp.blueyonder.co.uk/2006/05/klein-2-geometry.html". It may help folks who are looking for intuition:

Any vector space gives a 2-group - in fact a "2-vector space" of the sort http://www.arxiv.org/abs/math.QA.0307263" and I studied. It's actually familiar from basic linear algebra.

When you're first learning about vectors, for example vectors in the plane, it's a bit confusing, because <em>first</em> the teacher says that a vector is a <em>point</em> in the plane, or equivalently an arrow from the origin to that point... but <em>then</em> they draw vector as an <em>arrow</em> going from one point in the plane to another.

If you've ever taught linear algebra, you'll know that this issue leads to many confusions - especially when it's not explained clearly.

What's really going on here is that we're treating the plane as a 2-vector space, with a vector space of points (or objects) and a vector space of arrows between points (or morphisms).

You can compose arrows by sticking one at the end of the other, just like in any category. Namely: given arrows

p --v--> p'

and

p' --w--> p''

we stick them end to end and get an arrow

p --vow--> p''

People usually call this new arrow "v+w", but I'll write composition as "o", because we mustn't confuse it with another way to add arrows.

Namely: we can add points in the plane:

p + q

and we can also add arrows like this:

p --v--> p'

and

q --w--> q'

to get an arrow

p+q --v+w--> p'+q'

Get it? I wish I could draw pictures here... it's really simple stuff.

So, we have a category where the objects form a vector space, the morphisms form a vector space, and composition is a linear function from the vector space of composable pairs of arrows to the vector space of arrows! This is precisely a 2-vector space in the sense of Alissa and me (not to be confused with a Kapranov-Voevodsky 2-vector space).

Just as a vector space is a special sort of group, a 2-vector space is a special sort of 2-group!

So, we all meet a 2-group when we start learning about vectors, but we don't look it in the eye and see it for what it is.

 

[arivero]

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.

How does it work in the vector space example? Let's consider points A,B,C,D and morphisms A--f-->B, C--g-->D. We build a group of morphisms so that composition gf is a morphism from the point A+C to the point B+D.

Which is the neutral element of this group of morphisms?. It should be the morphism 0--e-->0 from the neutral element (of the group of points) to itself.

Now, given a morphism A--f-->B, which is the inverse morphism? It should be (-A)---f'--->(-B). Am I right?

 

[arivero]

Ok, perhaps the interesting question here is when does vow coincide with v+w. At a first glance, it seems that this condition let's one to find the neutral element of the group of points from a pure categorial setup.

(in the above example at least, this condition implies A=A+C, D=B+D and B=C, thus B=C=0)

 

[arivero]

The second idea that comes to the mind is to think if the additional structure can be used to exploit the convolution product (of functions from the morphisms to the field). Ie, we know that given two functions F(f), G(f) it is possible to build the function

<br /> Q(h)= \sum_{f \odot g=h} F(f) G(g)<br />

Now we could imagine a different thing
<br /> Q&#039;(h)= \sum_{f + g =h} F(f) G(g)<br />

Q and Q' coincide (do they?) when h is a morphism from the identity of the group of points to itself. In the above example this h is unique, because there is only one morphism between two given points. What does it happen in the general case?

 

[marcus]

from post #37 above

Here's a simple example of a 2-group taken from my discussions with http://www.dcorfield.pwp.blueyonder.co.uk/2006/05/klein-2-geometry.html".

...
...
p --vow--> p''

People usually call this new arrow "v+w", but I'll write composition as "o", because we mustn't confuse it with another way to add arrows.

Namely: we can add points in the plane:

p + q

and we can also add arrows like this:

p --v--> p'

and

q --w--> q'

to get an arrow

p+q --v+w--> p'+q'

Get it? I wish I could draw pictures here... it's really simple stuff.

...
...

I will confess that i don't get it.
I don't understand the "another way to add arrows", why it is different.
Oh yeah. maybe i do see it. different protocol.

you can compose morphisms if their heads and tails match, but
also you can ADD morphisms in such a way that you get a new source object and a new target
and to get the new source and the new target, you have to add the old sources and the old targets,
so then you have gone up one level

 

[Hurkyl]

Any vector space gives a 2-group - in fact a "2-vector space" of the sort Alissa Crans and I studied. It's actually familiar from basic linear algebra.

That is a cool 2-group. It's an irritating example, though, since it goes against my mental picture where a 2-group consists of a group of points, and each of those points in turn looks like a group.

On that note, you can turn your vector space into a 2-group in a different way! In fact, any Lie group becomes a 2-group whose 1-morphisms are points and 2-morphisms are tangent vectors. It seems easy enough to generalize to an arbitrary vector space.


I have a much harder time with the algebraic version -- e.g. AUT(G) for some group G. My brain rebels against the notion of the 2-morphisms as being translations (or automorphism+translations); it makes me feel like the 2-morphisms aren't higher than the 1-morphisms after all.

But now I wonder if that's simply an artifact of the fact that G can be viewed as if it was acting on itself?

 

[john baez]

pushing the Cauchy surface forwards

There's so much happening at once - it's hard to know what to do next.

That's why they invented space - so a bunch of things could happen at once. This leads to the problem of time: it's hard to know what to do next, because there's not a unique way to push the spacelike slice forwards in spacetime. You push your portion forwards, and I'll do mine.

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.

Me too! It seemed to me like you just just burst onto the scene knowing everything about 2-groups that it took me years to learn...

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! I hope you're healthy by now...

 

[john baez]

The 2-group you see on every blackboard

How does it work in the vector space example? Let's consider points A,B,C,D and morphisms A--f-->B, C--g-->D. We build a group of morphisms so that composition gf is a morphism from the point A+C to the point B+D.

Actually, composition of morphisms works as usual for categories: we can compose a morphism

A --f--> B

with a morphism

B --g--> C

to get a morphism

A --fg--> C

(I prefer to write fg or f o g for this composite; other people use gf or g o f, but this is a matter of taste.)

In our example, A, B and C are a triangle of dots on a blackboard. f is the unique arrow from A to B, g is the unique arrow from B to C, and composing them we get the unique arrow fg from A to C. This is one of the things we schoolteachers call vector addition, and it confuses the kids because no addition of the points A, B, C is involved.

Since we're using multiplicative notation for composition in our category, let's use additive notation for the group operation in our 2-group.

So, addition of

A--f-->B

and

C--g-->D

gives

A+C --f+g--> B+D

We add the coordinates of the points A and C, add the coordinates of B and D, and add each point on the unique arrow from A to C to the corresponding point on the unique arrow from B to D to get the unique arrow from A+C to B+D. This is something we schoolteachers don't often discuss! But it's perfectly sensible.

We build a group of morphisms so that composition gf is a morphism from the point A+C to the point B+D.

I'll be much happier if we use + everywhere for the 2-group operation, and say:


We build a group of morphisms so that the sum f+g is a morphism from the point A+C to the point B+D.

Which is the neutral element of this group of morphisms?. It should be the morphism 0--e-->0 from the neutral element (of the group of points) to itself.

Right! And, it can't hurt too much to use the name "0" for what you're calling e - it's the unique arrow from the origin to the origin, an arrow of 0 length that just sits at 0. Of course it's different from the object 0, which is the neutral element in the group of objects.

Now, given a morphism A--f-->B, which is the inverse morphism? It should be (-A)---f'--->(-B). Am I right?

Let's not mix up inverse for composition with inverse for addition; let's call the latter one the "negative".

Given the unique arrow from the point A on the blackboard to the point B:

A --f--> B

its inverse for composition is the unique arrow from B to A; we can call this

B --(f^{-1})--> A

Its negative is the unique arrow from -A to -B; we can call this

(-A) -- (-f) --> (-B)

We schoolteachers often use "-f" to stand for both these arrows, since they're the same length and point in the same direction... but we do this just to confuse the kiddies : they're different arrows, because they start at different points!

 

[john baez]

When your brain rebels

That is a cool 2-group. It's an irritating example, though, since it goes against my mental picture where a 2-group consists of a group of points, and each of those points in turn looks like a group.

Actually this means it's a great example, because you shouldn't think of each object in a 2-group as looking like a group - instead, it's the set of all objects which forms a group. Here it's the set of points in our vector space.

On that note, you can turn your vector space into a 2-group in a different way! In fact, any Lie group becomes a 2-group whose 1-morphisms are points and 2-morphisms are tangent vectors.

Yeah! That's the tangent 2-group of a Lie group. I like it because it's an example of a category where the morphisms actually are arrows. People draw morphisms as arrows, but here we're taking that seriously.

In the tangent 2-group the morphisms are tangent vectors, which go from a point in the Lie group to itself - they're "infinitesimal" arrows.

There's another 2-group built from a group where the objects are group elements and a morphism f: g -> h is a group element f with gf = h. You can think of f as a "finite-length" arrow going from g to h. As a special case, when our group is a vector space, we get the 2-vector space I've been talking about on this thread.

I have a much harder time with the algebraic version -- e.g. AUT(G) for some group G. My brain rebels against the notion of the 2-morphisms as being translations (or automorphism+translations); it makes me feel like the 2-morphisms aren't higher than the 1-morphisms after all.

I know what you mean. But don't let your brain rebel: who's in control, anyway - you or your brain? Hmm... that's actually a tough question.

The way to think of AUT(G) is that the objects are automorphisms

f: G -> G

and these form a group. But, there is an obvious thing that can go from one automorphism to another! Namely, we can conjugate an automorphism f by a group element g and get another automorphism f':

f&#039;(h) = gf(h)g^{-1} for all h in G

So, we take these "conjugations" as morphisms.

Of course the super-slick way to say this is that a group is a category, and AUT(G) is the 2-group with invertible functors f: G -> G as objects, and natural isomorphisms between these as morphisms... we can define this for any category G, not just a group.

But if you're more of a geometer than an algebraist, perhaps this will make your brain rebel.

 

[arivero]

but we do this just to confuse the kiddies :

Me? Well, true, devil we are.

Now, the third idea that comes to mind is that in this example of 2-group there is a well-known construction to get the tangent bundle as a limit. We tensor arrows times a open segment, say (0,1), so that now any arrow is labeled by the extreme points AND an epsilon in this segment; composition being as usual (A,B,e)o(B,C,e)=(A,C,e). The tangent bundle is pasted by defining, in the usual system of coordinates, that a sequence of arrows (A_n,B_n,e_n) going to (A,A,0) really converges towards the element (A, {B_n-A_n \over e_n}) of the tangent bundle.

Composition of morphisms survives in this limit, and one gets the expected composition on the tangent bundle (A,v)o(A,w)=(A,v+w). Here v,w are vectors in the tangent space T_AM

Then the question is, what does it happen with addition of morphisms? I'd say we get that (A,v)+(B,w)=(A+B,v+w). Very puzzling.

 

[marcus]

reaction to post #45, please ignore if too far off track

Of course the super-slick way to say this is that a group is a category, and AUT(G) is the 2-group with invertible functors f: G -> G as objects, and natural isomorphisms between these as morphisms... we can define this for any category G, not just a group.
...

what I am picturing seems slightly askew from this verbal description. maybe i can align my picture better, or find out what I'm doing wrong.
I have a group G that I view as a category with one object *
and what used to be "group elements" are now morphisms going * to *.

I want to picture AUT(G) as a twogroup. As you suggest, I can think of that as consisting of all the invertible functors from the category to itself, but why should I call the automorphisms "objects"? Maybe it would be more fun to take the category (embodying the group G) as my single OBJECT, and make the invertible functors my MORPHISMS (the first level of arrow-structure, always from and looping back to the same object) and then make the second level of arrow-structure be CONJUGATION BY GROUP ELEMENTS, since that will convert one morphism into another.

but that seems different from what you said. So I may have taken a false turn---may run into difficulty trying to verify the exchange rule.

 

[john baez]

level-shifting tricks

reaction to post #45, please ignore if too far off track
what I am picturing seems slightly askew from this verbal description. maybe i can align my picture better, or find out what I'm doing wrong.
I have a group G that I view as a category with one object *
and what used to be "group elements" are now morphisms going * to *.

Okay, you're viewing a group as a category with one object.

I want to picture AUT(G) as a twogroup. As you suggest, I can think of that as consisting of all the invertible functors from the category to itself, but why should I call the automorphisms "objects"?

Why not? (Less flippant answer follows.)

Maybe it would be more fun to take the category (embodying the group G) as my single OBJECT, and make the invertible functors my MORPHISMS (the first level of arrow-structure, always from and looping back to the same object) and then make the second level of arrow-structure be CONJUGATION BY GROUP ELEMENTS, since that will convert one morphism into another.

Okay - if you call these "conjugation by group element" guys 2-MORPHISMS, then you'll be thinking of the 2-group AUT(G) as a 2-category with one object.

That's fine if you like 2-categories.

But, since I tend to assume most people can just barely tolerate categories, much less 2-categories, I've been pushing a more lowbrow approach in this thread. I'm thinking of a 2-group as a category equipped with an extra operation which let's us "add" or "multiply" objects, and ditto for morphisms. I've said this before.

but that seems different from what you said.

No, the two approaches are equivalent. I mentioned this a while back:

I said this a while ago:

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!

In short:

LOWBROW: a group is a set with product, unit and inverses.

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

TRANSLATION: what we call elements in the lowbrow approach, are called morphisms in the highbrow approach.

LOWBROW: a 2-group is a category with product, unit and inverses.

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

TRANSLATION: what we call objects in the lowbrow approach, are called morphisms in the highbrow approach. What we call morphisms in the lowbrow approach, are called 2-morphisms in the highbrow approach.

I'm taking the low road. For example, in my "2-vector space you can draw on a blackboard", I'm saying we've got a category with objects being dots on the blackboard and morphisms being arrows. This category has a product, which we call addition - you can add dots, and you can add arrows. It also has a unit (a zero dot, and a zero arrow), and inverses (the negative of any dot, and the negative of any arrow).

But, we could also think of this as a 2-category with one object if we want. Then the dots get called morphisms and the arrows get called 2-morphisms.

I would never have brought this up, had you not forced me to.

This trick is an example of "level-shifting", which we do all the time in n-category theory. One man's first floor is another man's second floor! In fact, this also happens when you go from Europe to America: what they call the first floor in Europe is called the second floor in America.

In short, you're doing fine - you're just being American about it.

 

[john baez]

yeah, you see it

from post #37 above

I will confess that i don't get it.
I don't understand the "another way to add arrows", why it is different.
Oh yeah. maybe i do see it. different protocol.

you can compose morphisms if their heads and tails match, but
also you can ADD morphisms in such a way that you get a new source object and a new target
and to get the new source and the new target, you have to add the old sources and the old targets...

Yeah, you see it! If you draw an arrow in the plane from one point to another

(2,3) ---> (4,7)

and then you draw another arrow

(1,2) ---> (6,1)

you can add everything in sight and get a new arrow

(3,5) ---> (10,8)

Schoolteachers don't talk about this operation much, but if you teach vectors to kids, and tell them to "add" the above vectors, there's a good chance they'll do just this. They have to work to learn that what you meant by "addition" was to slide one arrow so its tail starts at the other's head:

(2,3) ---> (4,7) ---> (9,6)

and then compose them:

(2,4) ---> (9,6)

This is one reason kids have trouble with this stuff! We're talking about vector spaces, but our pictures of arrows are really all about 2-vector spaces!

So now you're struggling to unlearn that confusion...

 

[CarlB]

Frightening. I actually understood these last few posts. Despite having very little faith that this has anything to do with physics, and only clicking on the page so that it doesn't show up as a "new post" indicator on physics forums.

Carl

 

[Kea]

CarlB, can we tempt you to read a little more? Maybe some stuff about
quantum mechanical logic in diagrams?

Good! I hope you're healthy by now...

Yes, thank you John. I have started taking the big puppy for long walks in the bush.

I really like thinking of ordinary vector spaces this way. And soon perhaps we can teach the kids about them this way. One of my little nephews would probably refuse to do it any other way, because pictures make a lot more sense to him than lists of random looking rules.

The thing I really like (sorry to be so repetitive) about the 2-dimensional picture is that one day we can try and do Gray compositions of pieces of surfaces to make 3-dimensional pictures...and we can do this
sort of thing for ordinary vector spaces...which are like categorified numbers!

Well, it might be better to think of numbers (or polynomials) as tangles in a Riemann surface...but this is off topic...except that then by turning them into vector spaces we get things like sheaves! So it doesn't really matter how we try and do things...maybe we like doing String theory with Hecke eigensheaves...everything seems to end up at the same place at the end. It's always exciting to see that happening.

 

[CarlB]

Well the one thing that this sort of reminds me of is David Hestenes' comments on the Cambridge Geometric Algebra Group's gauge theory of gravity (GTG).

The comment was that it was significant that the theory could be put onto flat space, and Hestenes' reason for why this was something needed in the context of his "geometric algebra" seems to resonate with these ideas about vectors, particularly section IX, pages 21-23 of this link (which pages may be read without reading the rest of the paper):

Spacetime Geometry with Geometric Calculus
David Hestenes, To be published in the Preceedings of the Seventh International Conference on Clifford Algebra
http://modelingnts.la.asu.edu/pdf/SpacetimeGeometry.w.GC.proc.pdf

I'd give a brief description of the argument, but I don't think I can do it justice. Hetsenes does it so well and so clearly that I wouldn't want to butcher it by reducing its length and two pages is too long. Okay, but basically, the idea has to do with how one connects up an algebra to a manifold in such a way that one can do calculus on it.

For more on the geometric gauge theory of gravity, see the Cambridge geometric algebra group:
http://www.mrao.cam.ac.uk/~clifford/index.html

Carl

 

[Hurkyl]

Namely, we can conjugate an automorphism f by a group element g and get another automorphism f':

That, actually, is where I was getting stuck!

The problem is that something like R^n comes equipped with a mental image -- it's a space of points! But, my mental image of 2-automorphisms is very much like a homotopy of maps. While that generally works fine for natural transformations, I can't get it to mesh with my picture of R^n.

And to make things even more confusing... the 2-automorphisms of the vector space R^n look like the 1-automorphisms of the affine space R^n.

I think I'm okay if I pretend I don't know what R^n is, and just picture it as a dot with a bunch of loops hooked up to it... but I really don't think that's the right way to approach this problem.

 

[marcus]

JB began the thread with mention of a paper in the works by Baratin and Freidel-----extending to 4D what they have already done in 3D.

I think this other paper may be relevant. It just posted today and is also by Freidel, but with Starodubtsev and Kowalski-Glikman


http://arxiv.org/abs/gr-qc/0607014
Particles as Wilson lines of gravitational field
L. Freidel, J. Kowalski--Glikman, A. Starodubtsev
19 pages

"Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom."

 

[garrett]

I just read that new paper.

As far as I can tell, it works out exactly as you would expect point particles to behave in MacDowell-Mansouri BeeF gravity. Judging from their introduction, they seem oddly excited about it though, so maybe I'm missing something. It could be the excitement stems from their description of these particles as field monopoles, but I'm not sure why that's so different than putting the point particle actions in by hand. Anyway, it's a decent treatment and I like the approach.

 

[marcus]

I just read that new paper...

I am glad you had a look at it.
In a recent post, Baez mentioned that Freidel has 3 papers in the works with Starodubtsev and one in the work with Baratin. even if all don't come to fruition I'm inclined to expect at least a couple more in this same line of investigation.

It seems to me that you are especially well prepared to understand and comment, not only on this one but on the others when they come out.

The conclusions section speaks of a "forthcoming paper" in which they do a perturbation expansion in alpha

and thru that, they say, address the question of the flat limit of gravity and particles. I will get the quote

==quote==
First, since the alpha parameter is small, we can consider a perturbation theory of gravity coupled to particle(s) being the perturbation theory in alpha. The distinguished feature of this theory would be that it is, contrary to earlier approaches, manifestly diffeomorphism-invariant, so its framework it is possible to talk about weak gravitational field in the conceptual framework of full general relativity. These investigations, both in the case of beta = 0 and beta not = 0 will be presented in the forthcoming paper. The fuller control over the small alpha sector will presumably make it possible to address the outstanding question of what is the flat space limit of the theory of gravity, coupled to point particles. It has been claimed that such a theory will be not the special relativity, but some form of doubly special relativity
==endquote==

If they can show that the flatspace limit is not usual Lorentz but is, instead, some DSR, this would probably open up some possibilities to TEST. It would seem to me like considerable progress just to get a good flatspace limit of one sort or another.

 

[marcus]

I am mulling over this parameter alpha that they want to do the perturbation expansion in.
I think it came up in the earlier (Jan 2005?) Freidel Staro paper.
You see it on page 3 of this paper, equation 2.3

if alpha and beta were both zero then S would be a usual BF action, but alpha perturbs it and makes it deviate from the usual BF action. Am I wrong?

the nice thing is that we are now looking at a perturbation theory where we DO NOT HAVE A FIXED BACKGROUND GEOMETRY around which we perturb. I don't claim to have much grasp of this, but we seem to be contemplating the opportunity to "perturb around pure BeeF itself"

so they hold out the attractive notion of a background independent perturbation theory or I guess what they said was a "manifestly diffeomorphism invariant" perturbation theory. that was what they said in conclusions on page 15.

right now it looks to me as if they are proceeding with exactly what they promised in http://arxiv.org/hep-th/0501191 that they would do. rather than us getting new signals this time we are getting confirmation of progress along lines they said in january last year. Am I missing something?

 

[garrett]

Yep, that all sounds right.

The \alpha term is what makes BF into gravity. With \alpha itself proportional to the gravitational constant. Rovelli wrote about this as well, in his propagator paper. And, urr, I do BF too -- although I came to it rather circuitously.

"BF, it's what's for dinner."

 

[marcus]

... And, urr, I do BF too -- although I came to it rather circuitously.

Yes! and I am looking for you to surf this BF wave!

there is much truth in the saying
"BF, it's what's for dinner." I may adopt it as a signature.

 

[selfAdjoint]

there is much truth in the saying
"BF, it's what's for dinner." I may adopt it as a signature.

Although Prof. Baez once remarked that it should be "EF" which spoils the pun. He said the so-called B part of BF theory was not in fact like magnetism (which B traditionally expresses) but like electricity, E.