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: Code (Text): p p' \ / \ / \ / | | | p" 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
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/activi...ries/alltalks.cfm?CurrentPage=1&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
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.)
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)
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
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.co.uk/2006/05/changing-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!
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.co.uk/2006/05/changing-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.
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
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."
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
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."
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
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.
Greg Egan's questions 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.
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.
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.
It's not like BF theory is some alternative challenger. It has been shown that GR can be written as a BF theory with some extra stuff in the Lagrangian. So just as some people study lower dimensional models because they're easier, so others study BF theory because it's a simple theory that is "a lot like" GR and they know how to extend it to actually be GR.