A key sentence in TWF #232

marcus

this may have been emphasized at Usenet SPR, I didnt check. I will flag it here in case anyone overlooked it.
=====quote Baez TWF #232====

Second, suppose we let two particles collide and form a new one:
Now our worldlines don't form a submanifold anymore, but if we keep our wits about us, we can see that everything still makes sense, and we get momentum conservation in this form:
exp(p") = exp(p) exp(p')

since little loops going around the two incoming particles can fuse to form a loop going around the outgoing particle. Note that we're getting conservation of the group-valued momentum, not the Lie-algebra-valued momentum - we don't have

p" = p + p'

So, conservation of energy-momentum is getting modified by gravitational effects! This goes by the name of "doubly special relativity"...
====end quote====

I will see if the paste version copies OK

marcus

this comes in
http://math.ucr.edu/home/baez/week232.html
pretty far down the page

more discussion currently here
http://math.ucr.edu/home/baez/README.html
which currently describes Baez seminar talk

http://perimeterinstitute.ca/activit...&SeminarID=749

and colloquium at Perimeter
http://math.ucr.edu/home/baez/quantum_spacetime/

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

the reason I wanted to highlight that passage from #232 is that
the fundamental law of momentum conservation is a MULTIPLICATION of group elements
and it is only ADDITIVE TO FIRST ORDER

so what Newton told us about conservation of momentum, that it was an additive (Lie algebra) thing---that is only a first order approximation of the real multiplicative (Lie group) conservation law

or so this example suggests might be, which is kind of odd, and something I guess we should know about

john baez

Yeah, that's why I keep telling everyone about it!

However, I'm also trying to make it clear that only in 3d spacetime can we think of the momentum of a point particle as Lie-algebra-valued. So only in this case can we switch to using group-valued momentum.

That's because in 3 dimensions, Minkowski spacetime is a Lie algebra!

You first get an inkling of this when you learn about the "cross product" in linear algebra. The cross product is a special way to make 3d space into a Lie algebra, which doesn't work in other dimensions.

A similar trick works for 3d spacetime....

But for 4d spacetime, we need a different trick! That's what my papers with Crans, Wise and Perez are about.

We need to switch from point particles to strings! A string is just the right thing to have a momentum density valued in a Lie algebra, so we again can - and must! - switch to a group when we turn on gravity.

(Topological gravity, that is - in other words, BF theory. I'd like to get this to work for full-fledged gravity, but 4d gravity is, of course, tough.)

f-h

So Professor Baez, you're back in physics now?

marcus

you know I just checked in and saw post #3 and it seemed
to me that it was rather the best possible sort of mathematics

that is, inventing something new in the realm of logic and formal ideas that physicists can USE if they want to, and also people can just have fun with if they don't want to

not a thought out reaction on my part. but it seems to be a fit of sleepwalking mathematical inventiveness that comes at the right time to let some physicist fly get out of his bottle if he wants to---and he can stay in if he doesnt. (I mean the inventiveness you need to raise the show to 4D)

arivero

I wonder if it is still a quotient space in the sense of Connes example 2.beta, in pages 91-93 of the red book

selfAdjoint

Have you seen that David Corfiel posted a discussion between himself and Professor Baez on this and other topics at his blog: http://www.dcorfield.pwp.blueyonder....nging-rig.html.

Just a couple of mathematicians noodling around with the ideas that fascinate them, but how enlightening! And we have it here too! Wow!

selfAdjoint

Have you seen that David Corfiel post a discussion between himself and Professor Baez on this and other topics at his blog: http://www.dcorfield.pwp.blueyonder....nging-rig.html.

Just a couple of mathematicians noodling around with the ideas that fascinate them, but how enlightening! And we have it here too! Wow!

Also check out the stuff at TWF 233.

marcus

that's a good find. I had not found David Corfield's blog yet. thanks!

the conversation is long and rich in intuition---as far as I could tell from the initials it is JB talking much of the time.

not to actually "give up" on TWF 233, I should say that I have checked it out and find it not the easiest---I get more out of 232 and also little chunks of this Corfield blog. So I will print out the corfield and go read it in a more comfortable chair

selfAdjoint

Vector Analysis! They even satisfy the Jacobi Identity!

I remember the instructer trying to explain this to me in a course in Lie Algebras my first year of grad school. "But I was one and twenty; no use to talk to me."

marcus

housman is technically good, but the despair bothers me
we are practically in the cockpit with something very interesting, aren't we?

the focus of my confusion---where I completely say DUH! right now---is this

i understand that you can have a worldline in 3D and it can be a conical singularity persisting thru time and the deficit angle can be the mass

now he says that to generalize this to 4D we need a "world-pipe" or a "world-hose"-----which should be a VERY familiar idea to many of us
and this is swept out by a ring
and instead of mass of a point you have TENSION of the ring

and this makes something like a whole sheet of conical singularity persisting thru time

and this Y diagram that we have in this thread becomes a "Y-pipe"

this is what I want to think about before taking an afternoon nap. It was a nice holiday weekend here. hope all's well in the Midwest
=============

BTW just today, in Utrecht, Alejandro Perez was giving a talk to Loll's seminar exactly about this stuff:
quantization of branes coupled to beef-----talk had the same title as the Baez/Perez paper

selfAdjoint

Sunny and hot. Finshed spading the garden and watched my daughter planting perennials. And yes, thinking of braid groups and 2-categories and all that stuff because I am persuaded that the standard model and GR are both effective theories according to renormalization group logic and below them (at much higher energies) the physics may be even more complicated than what we have seen so far - why should we expect it to be simpler? So categorification, the capsulization of today's complexity in simpler forms which can interact to become tomorrow's complexity. So I've got to get my head around this stuff that Kea and Professor Baez and Corfield and yes, Urs Scheiber have been pushing....

As Feynmann said in a slightly different context; "There's plenty of room at the bottom."

marcus

yes good for Urs! I saw his comment after Corfield's discussion with Baez. And his comment on the "loop braid" paper(s) at Coffee Table.

being in the garden these days gives me an idea that Category theory is historically like the appearance of the flowering plants. All it does really is SPEED UP THE RATE OF MATHEMATICAL INVENTION.

you just set up a new mechanism, namely insects that fly around, or you set up a functors to be the embodiment of analogy, and it speeds up the rate of evolution.

in the end, perhaps it could have happened the old slow way too, maybe

and then there are people such as Kea who I suspect just like the flowers

Kea

Oh, Marcus, thank you, that made my day. Hi, selfAdjoint! Yes, Corfield has a nice blog. Plenty of room at the bottom and all that.

john baez

Greg Egan had some nice questions about this "group-valued momentum" business, which I was able to answer in part. However, I think there's still some work to be done to dig out the full meaning of this concept - some calculations, thought experiments, and so on.

Egan wrote:

John Baez wrote:

> The really cool part is the relation between the Lie algebra
> element p and the group element exp(p). Originally we thought
> of p as momentum - but there's a sense in which exp(p) is the
> momentum that really counts!

Would it be correct to assume that the ordinary tangent vector p still
transforms in the usual way? In other words, suppose I'm living in a
2+1 dimensional universe, and there's a point particle with rest mass m
and hence energy-momentum vector in its rest frame of p=m e_0. If I
cross its world line with a certain relative velocity, there's an
element g of SO(2,1) which tells me how to map the particle's tangent
space to my own. Would I measure the particle's energy-momentum to be
p'=gp? (e.g. if I used the particle to do work in my own rest frame)
Would there still be no upper bound on the total energy, i.e. by making
our relative velocity close enough to c, I could measure the particle's
kinetic energy to be as high as I wished?

I guess I'm trying to clarify whether the usual Lorentz transformation
of the tangent space has somehow been completely invalidated for
extreme boosts, or whether it's just a matter of there being a second
definition of "momentum" (defined in terms of the Hamiltonian) which
transforms differently and is the appropriate thing to consider in
gravitational contexts.

In other words, does the cut-off mass apply only to the deficit angle,
and do boosts still allow me to measure (by non-gravitational means)
arbitrarily large energies (at least in the classical theory)?

I replied:

Greg Egan wrote:

>John Baez wrote:

>>The really cool part is the relation between the Lie algebra
>>element p and the group element exp(p). Originally we thought
>>of p as momentum - but there's a sense in which exp(p) is the
>>momentum that really counts!

>Would it be correct to assume that the ordinary tangent vector p
>still transforms in the usual way?

Hi! Yes, it would.

>In other words, suppose I'm living in a 2+1 dimensional universe,
>and there's a point particle with rest mass m and hence
>energy-momentum vector in its rest frame of p=m e_0. If I
>cross its world line with a certain relative velocity, there's
>an element g of SO(2,1) which tells me how to map the particle's
>tangent space to my own. Would I measure the particle's
>energy-momentum to be p'=gp? (e.g. if I used the particle to
>do work in my own rest frame) Would there still be no upper
>bound on the total energy, i.e. by making our relative velocity
>close enough to c, I could measure the particle's kinetic energy
>to be as high as I wished?

To understand this, it's good to think of the momenta as
elements of the Lie algebra so(2,1) - it's crucial to the
game.

Then, if you have momentum p, and I zip past you, so you
appear transformed by some element g of the Lorentz group
SO(2,1), I'll see your momentum as

p' = g p g^{-1}

This is just another way of writing the usual formula for
Lorentz transforms in 3d Minkowski space. No new physics
so far, just a clever mathematical formalism.

But when we turn on gravity, letting Newton's constant k
be nonzero, we should instead think of momentum as group-valued,
via

h = exp(kp)

and similarly

h' = exp(kp')

Different choices of p now map to the same choice of h.
In particular, a particle of a certain large mass - the
Planck mass- will turn out to act just like a particle
of zero mass!

So, if we agree to work with h instead of p, we are now
doing new physics. This is even more obvious when we decide
to multiply momenta instead of adding them, since multiplication
in SO(2,1) is noncommutative!

But, if we transform our group-valued momentum in the correct
way:

h' = ghg^{-1}

this will be completely compatible with our previous transformation
law for vector-valued momentum!

>I guess I'm trying to clarify whether the usual Lorentz transformation
>of the tangent space has somehow been completely invalidated for
>extreme boosts, or whether it's just a matter of there being a second
>definition of "momentum" (defined in terms of the Hamiltonian) which
>transforms differently and is the appropriate thing to consider in
>gravitational contexts.

Good question! Amazingly, the usual Lorentz transformations still
work EXACTLY - even though the rule for adding momentum is new (now
it's multiplication in the group). We're just taking exp(kp) instead
of p as the "physical" aspect of momentum.

This effectively puts an upper limit on mass, since as
we keep increasing the mass of a particle, eventually it "loops
around" SO(2,1) and act exactly like a particle of zero mass.

But, it doesn't exactly put an upper bound on energy-momentum,
since SO(2,1) is noncompact. Of course energy and momentum don't
take real values anymore, so one must be a bit careful with this
"upper bound" talk.

>In other words, does the cut-off mass apply only to the deficit
>angle, and do boosts still allow me to measure (by non-gravitational
>means) arbitrarily large energies (at least in the classical theory)?

There's some sense in which energy-momenta can be arbitrarily
large. That's because the space of energy-momenta, namely SO(2,1),
is noncompact. Maybe you can figure out some more intuitive way
to express this.

john baez

I don't know. Maybe I'm neither in nor out. Maybe I'm in a momentum eigenstate, not a position eigenstate.

Most of the time I really want to do math, stuff involving n-categories and the like. But I have these grad students, Jeffrey Morton and Derek Wise, who are eager to work on quantum gravity. So, we had to dream up a project for them to work on, and this "strings in 4d BF theory" was what we cooked up.

If all goes well, this will hook up to Freidel and Starodubtsev's work on treating 4d gravity as a BF theory plus a perturbation term, which is supposed to lead to a spin foam model for quantum gravity. In fact the G = 0 version of this spin foam model seems to be related to some work of Kea's! Then, with more luck, we will see strings and also particles showing up in this spin foam model. (I was an idiot, I saw where the strings fit in but not the particles! - Freidel pointed that out after Baratin's talk last week. But, if I'd seen the particles, I might not have seen the strings.)

Dreams, dreams...

It might work; it probably won't.

But, luckily, all that really matters is Jeff and Derek will have nice theses. If I think about it this way, it takes the pressure off, and I can just have fun. It's important to have fun.

Chronos

Why BF? There are other approaches that appear equally promising to me. That took me by surprise - I didn't know you favored BF. PS, I really like category theory.