A key sentence in TWF #232

  1. marcus

    marcus 25,079
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    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
     
    Last edited: May 28, 2006
  2. jcsd
  3. marcus

    marcus 25,079
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    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
     
  4. john baez

    john baez 178
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    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.)
     
  5. So Professor Baez, you're back in physics now?
     
  6. marcus

    marcus 25,079
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    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)
     
    Last edited: May 29, 2006
  7. arivero

    arivero 3,036
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    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
     
  8. selfAdjoint

    selfAdjoint 8,147
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    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!
     
  9. selfAdjoint

    selfAdjoint 8,147
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    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.
     
  10. marcus

    marcus 25,079
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    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
     
    Last edited: May 29, 2006
  11. selfAdjoint

    selfAdjoint 8,147
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    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."
     
  12. marcus

    marcus 25,079
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    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:smile:
    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
     
    Last edited: May 29, 2006
  13. selfAdjoint

    selfAdjoint 8,147
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    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."
     
  14. marcus

    marcus 25,079
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    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
     
    Last edited: May 29, 2006
  15. Oh, Marcus, thank you, that made my day. :smile: Hi, selfAdjoint! Yes, Corfield has a nice blog. Plenty of room at the bottom and all that.
     
    Last edited: May 29, 2006
  16. john baez

    john baez 178
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    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.​
     
  17. john baez

    john baez 178
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    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. :wink:
     
    Last edited: May 29, 2006
  18. Chronos

    Chronos 10,217
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    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.
     
    Last edited: May 30, 2006
  19. selfAdjoint

    selfAdjoint 8,147
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    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.
     
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