Hi Folks: Due to a spam attack that has lasted for several hours now comments are currently disabled. Sorry about that. I will try to change settings in the afternoon to see whether the storm has passed by.
Well, with no particular topic posted may I ask here: Does Noether's Theorem generalize to universes analogous to ours but with other number of space dimensions? It seems like it should, but I've seen various statements that action etc. has special properties in three-D space.
Well, some comments are challenged even when not disabled.
Neil: Yes. Noether's theorem has nothing to do with the number of dimensions. What matters is the symmetry of the system (or its absence). If you e.g. have a spacetime that is not translation invariant to begin with (because some particular submanifold is 'special', for example because we sit on it) you have no reason to expect conserved charges.
I am not sure what 'special properties' you refer to. The only thing that springs to my mind that's special about the action in 3+1 dimension is the dimensionality of the coupling constants. That however is completely irrelevant for Noether's theorem. Best,
Thanks, Bee. I don't get all that as a middle-brow but I've seen quotes like, "In three dimensions, a vector is dual to a scalar. This means we can dualize all vectors in our three-dimensional action and obtain an action that only consists of scalar fields and a metrical field." But I think related to the above but more interesting: only in three dimensions can angular momentum be represented as a vector (the rotational plane would have no perp. in 2-D, be perp. to a plane in 3-D, to a space in 4-D etc. and only to a vector ("axis") in 3-D.
I don't think that foils the attempt to show how symmetry brings forth conservation, but angular momentum can't be treated as a vector in those other spaces so has to be represented differently. Also, there are more "ways for it to point" in higher spaces, than for a vector. For example in 2-D AM is just a scalar, in 4-D it has six fundamental ways to orient rather than four (given x,y,z,w; we have fundamental unit planes xy, xz, xw, yz, yw, zw) and so on as D(D-1)/2.
Uncle Al said around here that NT had a loophole allowing for non-conservation of angular momentum, I don't know how valid that is.
That's a different issue. You were asking whether Noether's theorem generalizes to N dimensions. Yes, it does. What you are taking about now is the question what symmetries you can have in N dimensions.
I have no clue what loophole in Noether's theorem Plato thinks he has discovered.
http://www.cnn.com/2009/TECH/ptech/01/16/virus.downadup/index.html
ReplyDeleteMicrocap - when the worst is good enough.
The politically correct term is "challenged", not "disabled".
ReplyDeleteHi Arun,
ReplyDelete“The politically correct term is "challenged", not "disabled".”
In the context it’s usually applied the challenge is on going and therefore let’s hope it doesn’t qualify as being so defined.
Best,
Phil
Well, with no particular topic posted may I ask here: Does Noether's Theorem generalize to universes analogous to ours but with other number of space dimensions? It seems like it should, but I've seen various statements that action etc. has special properties in three-D space.
ReplyDeleteWell, some comments are challenged even when not disabled.
ReplyDeleteNeil: Yes. Noether's theorem has nothing to do with the number of dimensions. What matters is the symmetry of the system (or its absence). If you e.g. have a spacetime that is not translation invariant to begin with (because some particular submanifold is 'special', for example because we sit on it) you have no reason to expect conserved charges.
I am not sure what 'special properties' you refer to. The only thing that springs to my mind that's special about the action in 3+1 dimension is the dimensionality of the coupling constants. That however is completely irrelevant for Noether's theorem. Best,
B.
Oooh, I didn't mean to imply a connection between the first paragraph of my above comment and the following two.
ReplyDeleteThanks, Bee. I don't get all that as a middle-brow but I've seen quotes like, "In three dimensions, a vector is dual to a scalar. This means we can dualize all vectors in our three-dimensional action and obtain an action that only consists of scalar fields and a metrical field." But I think related to the above but more interesting: only in three dimensions can angular momentum be represented as a vector (the rotational plane would have no perp. in 2-D, be perp. to a plane in 3-D, to a space in 4-D etc. and only to a vector ("axis") in 3-D.
ReplyDeleteI don't think that foils the attempt to show how symmetry brings forth conservation, but angular momentum can't be treated as a vector in those other spaces so has to be represented differently. Also, there are more "ways for it to point" in higher spaces, than for a vector. For example in 2-D AM is just a scalar, in 4-D it has six fundamental ways to orient rather than four (given x,y,z,w; we have fundamental unit planes xy, xz, xw, yz, yw, zw) and so on as D(D-1)/2.
Uncle Al said around here that NT had a loophole allowing for non-conservation of angular momentum, I don't know how valid that is.
Hi Neil:
ReplyDeleteThat's a different issue. You were asking whether Noether's theorem generalizes to N dimensions. Yes, it does. What you are taking about now is the question what symmetries you can have in N dimensions.
I have no clue what loophole in Noether's theorem Plato thinks he has discovered.
Best,
B.
If your space has a loophole, presumably it is at the cost of a symmetry.....
ReplyDelete